Is barbecue allowed in van saun county park beside the playground area ?

Yes

Yes, although with nice weather, I might suggest getting there early.

Yes they have certain areas for BBQ

If your a small group it's fine but if your with a larger group you might want to check in with them to make sure.

We’re only 3 so I guess it’s fine

Yes, but you might need a permit if your group is big. Check their website for rules.