Representatives in Disjoint Set Union

Question

The goal is to efficiently obtain representatives of every set. Obtaining the root as representative is better since it gives more information about the set; nodes are often augmented with properties of their subtrees.

For example, given the sequence of operations

make_set(1)
make_set(2)
make_set(3)
union_sets(2,3)

resulting dsu

we want to efficiently obtain the parents <1, 2>.

A possible solution:

s = empty set
for (every node v):
  s.insert(find_parent(v))

There are n nodes, finding parent has constant amortized runtime, inserting into a set Theta(log n). The worst-case time is Theta(n log n). How could this solution be improved?

If disjoint-set union is ill-suited for this purpose, is there a better approach?

Edit: Matt Timmermans pointed out that finding parent takes constant amortized time. And that using the constant time

is_root(v):
  return v == parent[v]

we can also obtain Theta(n log n) worst-case time. Nonetheless, is there an augmentation that would allow for better runtime?