One of our favourite parks in Bangkok for cycling is Suan Rot Fai which is part of the bigger Chatuchak Park in northern Bangkok I think went on a Sunday morning which is the best time. It is open every day from 5 a.m. to 9 p.m. Though I don’t advise you to go here at night for safety reasons.
There are plenty of things for families to do in this 150 acre park. If you don’t have your own bikes, you can rent them for as little as 20 Baht from the shops at the northern end. There are also playgrounds for kids to amuse themselves. There is also a free Butterfly Garden that is open Tuesdays to Sundays. If you don’t want to bring your own food, there are plenty of stalls selling food in the same area where they rent out bicycles