My friend sons birthday is comming up and She has wanted to have a party in prospect with a bounce house, are bounce houses allowed?

Yes You can play music And a barbecue, too

I see people doing it but check out the websites to see if you may have to get permission

Hi.. Perhaps this site will help you.. It's the Prospect Park web site https://www.prospectpark.org/visit-the-park/general-info/permits/small-events/

I've never seen one in the park, but you can definitely have a cool party with barbeque. There might be some sort of permission required, though, so I'd recommend to check a website, hope this helps!