diff --git a/git-repository/src/revision/spec/parse/delegate.rs b/git-repository/src/revision/spec/parse/delegate.rs
index a4d7a876ad..e0ab65a479 100644
--- a/git-repository/src/revision/spec/parse/delegate.rs
+++ b/git-repository/src/revision/spec/parse/delegate.rs
@@ -286,9 +286,60 @@ impl<'repo> delegate::Revision for Delegate<'repo> {
 }
 
 impl<'repo> delegate::Navigate for Delegate<'repo> {
-    fn traverse(&mut self, _kind: Traversal) -> Option<()> {
+    fn traverse(&mut self, kind: Traversal) -> Option<()> {
         self.unset_disambiguate_call();
-        todo!()
+        self.follow_refs_to_objects_if_needed()?;
+
+        let mut replacements = SmallVec::<[(ObjectId, ObjectId); 1]>::default();
+        let mut errors = Vec::new();
+        let objs = self.objs[self.idx].as_mut()?;
+        let repo = self.repo;
+
+        for obj in objs.iter() {
+            match kind {
+                Traversal::NthParent(num) => {
+                    match self.repo.find_object(*obj).map_err(Error::from).and_then(|obj| {
+                        obj.try_into_commit().map_err(|err| {
+                            let object::try_into::Error { actual, expected, id } = err;
+                            Error::ObjectKind {
+                                oid: id.attach(repo).shorten_or_id(),
+                                actual,
+                                expected,
+                            }
+                        })
+                    }) {
+                        Ok(commit) => match commit.parent_ids().skip(num.saturating_sub(1)).next() {
+                            Some(id) => replacements.push((commit.id, id.detach())),
+                            None => errors.push((
+                                commit.id,
+                                Error::ParentOutOfRange {
+                                    oid: commit.id().shorten_or_id(),
+                                    desired: num,
+                                    available: commit.parent_ids().count(),
+                                },
+                            )),
+                        },
+                        Err(err) => errors.push((*obj, err)),
+                    }
+                }
+                Traversal::NthAncestor(_num) => todo!("ancestor"),
+            }
+        }
+
+        if errors.len() == objs.len() {
+            self.err.extend(errors.into_iter().map(|(_, err)| err));
+            None
+        } else {
+            for (obj, err) in errors {
+                objs.remove(&obj);
+                self.err.push(err);
+            }
+            for (find, replace) in replacements {
+                objs.remove(&find);
+                objs.insert(replace);
+            }
+            Some(())
+        }
     }
 
     fn peel_until(&mut self, kind: PeelTo<'_>) -> Option<()> {
diff --git a/git-repository/src/revision/spec/parse/types.rs b/git-repository/src/revision/spec/parse/types.rs
index 28c181bfb5..70b4871b85 100644
--- a/git-repository/src/revision/spec/parse/types.rs
+++ b/git-repository/src/revision/spec/parse/types.rs
@@ -61,6 +61,12 @@ pub struct Options {
 #[derive(Debug, thiserror::Error)]
 #[allow(missing_docs)]
 pub enum Error {
+    #[error("Commit {oid} has {available} parents and parent number {desired} is out of range")]
+    ParentOutOfRange {
+        oid: git_hash::Prefix,
+        desired: usize,
+        available: usize,
+    },
     #[error("Path {desired_path:?} did not exist in index at stage {desired_stage}{}{}", stage_hint.map(|actual|format!(". It does exist at stage {actual}")).unwrap_or_default(), exists.then(|| ". It exists on disk").unwrap_or(". It does not exist on disk"))]
     IndexLookup {
         desired_path: BString,
diff --git a/git-repository/tests/fixtures/generated-archives/make_rev_spec_parse_repos.tar.xz b/git-repository/tests/fixtures/generated-archives/make_rev_spec_parse_repos.tar.xz
index 4cade160c6..c1310f8e22 100644
--- a/git-repository/tests/fixtures/generated-archives/make_rev_spec_parse_repos.tar.xz
+++ b/git-repository/tests/fixtures/generated-archives/make_rev_spec_parse_repos.tar.xz
@@ -1,3 +1,3 @@
 version https://git-lfs.github.com/spec/v1
-oid sha256:e40c952da2d14b24fd94d1a47c7c753d634674145c938853af39a922453a7c57
-size 27956
+oid sha256:a6db0e8ed85ec2e7a3457d411a65a3508683de5156f45baf551c8d019b26e2e3
+size 28012
diff --git a/git-repository/tests/fixtures/make_rev_spec_parse_repos.sh b/git-repository/tests/fixtures/make_rev_spec_parse_repos.sh
index b108bdd662..46949e4574 100644
--- a/git-repository/tests/fixtures/make_rev_spec_parse_repos.sh
+++ b/git-repository/tests/fixtures/make_rev_spec_parse_repos.sh
@@ -336,6 +336,7 @@ git init complex_graph
   git merge c || :
   { echo g && echo h && echo i && echo j && echo d && echo e && echo f && echo b && echo c && echo a; } > file
   git add file && git commit -m A
+  git branch a
 
   baseline ":/message" # finds 'message recent' instead of 'initial message'
   baseline ":/!-message" # above, negated
@@ -346,4 +347,9 @@ git init complex_graph
   baseline ":file"      # index lookup, default stage 0
   baseline ":1:file"    # stage 1
   baseline ":foo"       # not found
+  # parents
+  baseline "a"
+  baseline "a^1"
+  baseline "a^0"
+  baseline "a^42"
 )
diff --git a/git-repository/tests/revision/spec/from_bytes/mod.rs b/git-repository/tests/revision/spec/from_bytes/mod.rs
index 96b13f5b59..390617cee9 100644
--- a/git-repository/tests/revision/spec/from_bytes/mod.rs
+++ b/git-repository/tests/revision/spec/from_bytes/mod.rs
@@ -7,6 +7,28 @@ pub use util::*;
 
 mod ambiguous;
 mod regex;
+
+mod traverse {
+    use crate::revision::spec::from_bytes::{parse_spec, repo};
+    use git_repository::prelude::ObjectIdExt;
+    use git_repository::revision::Spec;
+    use git_testtools::hex_to_id;
+
+    #[test]
+    fn parent() {
+        let repo = repo("complex_graph").unwrap();
+        assert_eq!(
+            parse_spec("a^1", &repo).unwrap(),
+            Spec::from_id(hex_to_id("5b3f9e24965d0b28780b7ce5daf2b5b7f7e0459f").attach(&repo))
+        );
+        assert_eq!(parse_spec("a", &repo).unwrap(), parse_spec("a^0", &repo).unwrap(),);
+        assert_eq!(
+            parse_spec("a^42", &repo).unwrap_err().to_string(),
+            "Commit 55e825e has 2 parents and parent number 42 is out of range"
+        );
+    }
+}
+
 mod index {
     use crate::revision::spec::from_bytes::{parse_spec, repo};
     use git_repository::prelude::ObjectIdExt;