Does "go dtí" causes the lenition here?
No; because páirc is feminine, an causes the lenition.
Would "Téann sé don pháirc" also be correct?
No. go (dtí), chuig, and chun can all be used to express direction, however. See here. Note that go dtí is used before the article, otherwise it's just go (Téann sé go páirc - He (habitually) goes to a park)
Okay; good to know. Thank you.
I didn't know about dtí only being used with an article, either, so that's also very helpful.