go, goes, going ?

Is this a correct sentence : “In her free time, she likes playing snooker, and go for a walk in the park.” Or does “go” need to be changed into “goes” (describing a habit) ?

Or should “go” be changed into “going” because of the parallelism rule (playing snooker, going for a walk) ?

Or are all three verbs “go”, “goes”, “going” correct ?

Only ‘going’ is correct in that sentence.

In her free time, she likes playing snooker and going for a walk in the park.

You would have to make slight changes for the other forms to be acceptable:
For example:
In her free time she likes playing snooker. She also likes to go for a walk in the park.
In her free time she plays snooker. Sometimes she goes for a walk in the park.