diff --git a/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-call/input.js b/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-call/input.js
new file mode 100644
index 000000000000..ad8de24b77d2
--- /dev/null
+++ b/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-call/input.js
@@ -0,0 +1,6 @@
+class Bar extends Foo {
+  constructor() {
+      let f = () => super();
+      f();
+  }
+}
diff --git a/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-call/options.json b/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-call/options.json
new file mode 100644
index 000000000000..eb924af09529
--- /dev/null
+++ b/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-call/options.json
@@ -0,0 +1,4 @@
+{
+  "plugins": ["transform-arrow-functions"],
+  "throws": "When using '@babel/plugin-transform-arrow-functions', it's not possible to compile `super()` in an arrow function without compiling classes.\nPlease add '@babel/plugin-transform-classes' to your Babel configuration."
+}
diff --git a/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-prop/input.js b/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-prop/input.js
new file mode 100644
index 000000000000..c3f597bc5ca2
--- /dev/null
+++ b/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-prop/input.js
@@ -0,0 +1,6 @@
+class Bar extends Foo {
+  constructor() {
+      let f = () => super.x;
+      f();
+  }
+}
diff --git a/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-prop/options.json b/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-prop/options.json
new file mode 100644
index 000000000000..f6a099e18a49
--- /dev/null
+++ b/packages/babel-plugin-transform-arrow-functions/test/fixtures/arrow-functions/super-prop/options.json
@@ -0,0 +1,4 @@
+{
+  "plugins": ["transform-arrow-functions"],
+  "throws": "When using '@babel/plugin-transform-arrow-functions', it's not possible to compile `super.prop` in an arrow function without compiling classes.\nPlease add '@babel/plugin-transform-classes' to your Babel configuration."
+}
diff --git a/packages/babel-plugin-transform-parameters/src/index.ts b/packages/babel-plugin-transform-parameters/src/index.ts
index a316a0033a09..d0ec02dd7cee 100644
--- a/packages/babel-plugin-transform-parameters/src/index.ts
+++ b/packages/babel-plugin-transform-parameters/src/index.ts
@@ -27,7 +27,10 @@ export default declare((api, options: Options) => {
             .some(param => param.isRestElement() || param.isAssignmentPattern())
         ) {
           // default/rest visitors require access to `arguments`, so it cannot be an arrow
-          path.arrowFunctionToExpression({ noNewArrows });
+          path.arrowFunctionToExpression({
+            allowInsertArrowWithRest: false,
+            noNewArrows,
+          });
 
           // In some cases arrowFunctionToExpression replaces the function with a wrapper.
           // Return early; the wrapped function will be visited later in the AST traversal.
diff --git a/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-default-super-no-classes-plugin/input.js b/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-default-super-no-classes-plugin/input.js
new file mode 100644
index 000000000000..b9a3692a568c
--- /dev/null
+++ b/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-default-super-no-classes-plugin/input.js
@@ -0,0 +1,6 @@
+class Bar extends Foo {
+    constructor() {
+        let f = (x = super()) => x;
+        f();
+    }
+}
diff --git a/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-default-super-no-classes-plugin/options.json b/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-default-super-no-classes-plugin/options.json
new file mode 100644
index 000000000000..78d5c966915c
--- /dev/null
+++ b/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-default-super-no-classes-plugin/options.json
@@ -0,0 +1,4 @@
+{
+    "plugins": ["transform-parameters"],
+    "throws": "When using '@babel/plugin-transform-parameters', it's not possible to compile `super()` in an arrow function with default or rest parameters without compiling classes.\nPlease add '@babel/plugin-transform-classes' to your Babel configuration."
+}
diff --git a/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-rest-super-no-classes-plugin/input.js b/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-rest-super-no-classes-plugin/input.js
new file mode 100644
index 000000000000..df404a5f5bb4
--- /dev/null
+++ b/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-rest-super-no-classes-plugin/input.js
@@ -0,0 +1,6 @@
+class Bar extends Foo {
+    constructor() {
+        let f = (...args) => super(...args);
+        f();
+    }
+}
diff --git a/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-rest-super-no-classes-plugin/options.json b/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-rest-super-no-classes-plugin/options.json
new file mode 100644
index 000000000000..78d5c966915c
--- /dev/null
+++ b/packages/babel-plugin-transform-parameters/test/fixtures/parameters/arrow-rest-super-no-classes-plugin/options.json
@@ -0,0 +1,4 @@
+{
+    "plugins": ["transform-parameters"],
+    "throws": "When using '@babel/plugin-transform-parameters', it's not possible to compile `super()` in an arrow function with default or rest parameters without compiling classes.\nPlease add '@babel/plugin-transform-classes' to your Babel configuration."
+}
diff --git a/packages/babel-traverse/src/path/conversion.ts b/packages/babel-traverse/src/path/conversion.ts
index eecdf85af83a..041abea0d8c1 100644
--- a/packages/babel-traverse/src/path/conversion.ts
+++ b/packages/babel-traverse/src/path/conversion.ts
@@ -149,12 +149,14 @@ export function arrowFunctionToExpression(
   this: NodePath<t.ArrowFunctionExpression>,
   {
     allowInsertArrow = true,
+    allowInsertArrowWithRest = allowInsertArrow,
     /** @deprecated Use `noNewArrows` instead */
     specCompliant = false,
     // TODO(Babel 8): Consider defaulting to `false` for spec compliancy
     noNewArrows = !specCompliant,
   }: {
     allowInsertArrow?: boolean | void;
+    allowInsertArrowWithRest?: boolean | void;
     specCompliant?: boolean | void;
     noNewArrows?: boolean;
   } = {},
@@ -169,6 +171,7 @@ export function arrowFunctionToExpression(
     this,
     noNewArrows,
     allowInsertArrow,
+    allowInsertArrowWithRest,
   );
 
   // @ts-expect-error TS requires explicit fn type annotation
@@ -240,6 +243,7 @@ function hoistFunctionEnvironment(
   // TODO(Babel 8): Consider defaulting to `false` for spec compliancy
   noNewArrows: boolean | void = true,
   allowInsertArrow: boolean | void = true,
+  allowInsertArrowWithRest: boolean | void = true,
 ): { thisBinding: string; fnPath: NodePath<t.Function> } {
   let arrowParent;
   let thisEnvFn: NodePath<t.Function> = fnPath.findParent(p => {
@@ -286,7 +290,17 @@ function hoistFunctionEnvironment(
   if (inConstructor && superCalls.length > 0) {
     if (!allowInsertArrow) {
       throw superCalls[0].buildCodeFrameError(
-        "Unable to handle nested super() usage in arrow",
+        "When using '@babel/plugin-transform-arrow-functions', " +
+          "it's not possible to compile `super()` in an arrow function without compiling classes.\n" +
+          "Please add '@babel/plugin-transform-classes' to your Babel configuration.",
+      );
+    }
+    if (!allowInsertArrowWithRest) {
+      // preset-env with target `since 2017` enables `transform-parameters` without `transform-classes`.
+      throw superCalls[0].buildCodeFrameError(
+        "When using '@babel/plugin-transform-parameters', " +
+          "it's not possible to compile `super()` in an arrow function with default or rest parameters without compiling classes.\n" +
+          "Please add '@babel/plugin-transform-classes' to your Babel configuration.",
       );
     }
     const allSuperCalls: NodePath<t.CallExpression>[] = [];
@@ -345,7 +359,9 @@ function hoistFunctionEnvironment(
   if (superProps.length > 0) {
     if (!allowInsertArrow) {
       throw superProps[0].buildCodeFrameError(
-        "Unable to handle nested super.prop usage",
+        "When using '@babel/plugin-transform-arrow-functions', " +
+          "it's not possible to compile `super.prop` in an arrow function without compiling classes.\n" +
+          "Please add '@babel/plugin-transform-classes' to your Babel configuration.",
       );
     }