I would say more simply, "After taking her to the park, I went home."

It's grammatically correct, but it's redundant: the time of "having taken" is already after the time of taking, so you are talking about a time after that again, with no reference to the intermediate time. So why not just say "Having taken her ..." or "After I took her ...", or if you like, "After taking her ..."?

I agree with Stephanie. After having + past participle may be acceptable but it really sounds formal and a bit clumsy. I don't recommend it. The other acceptable option is : "Having taken her, ..."

I would say "After taking her to the park, I went home" but both are fine.

It sounds very natural and not too formal to my US native ear.