Join a gym or community centre (like YMCA) that offers adult programs & childminding. This way, you can have some fun & interact w/adults & your child can do the same w/kids.
As for parks & dads, you need to get over that. Your child needs to have fun outside w/other kids. Noone needs to know you aren't married unless YOU mention it. Six years old is old enough for a bicycle, why don't you get one for yourself also & go riding together?
Second thoughts can generally be amended with judicious action; injudicious actions can seldom be recovered with second thoughts.
--Cyteen by C.J.Cherryh