Google is your friend. Why? Simply because the Google maps functionality will have to finding suitable parks in no time. You can look specifically ones that cater to cyclists, but if they don't then there's no need to stress. Most parents won't mind a child riding a bike and they might even bring their own child's bike with them the next time.
A Empty Piece Of Land.
There are many plots of land that just stand empty, so why not make use of them? All you need to do is a very small amount of scouting work prior to taking your child there. Watch out for rocks, gravel and rocky regions with no grass. If there are lots of shrubberies, then that would be perfect. If the child does happen to fall on the kid bike then the shrubberies will be there to protect them.
Take them back to school.
Now, your child might be too young to go to school yet, but most schools will allow you use their fields if you are in their area. After all, your child will most probably go tot hat school in the future and they would want to build up good will and school pride as early as possible. You can feel safe knowing that your child is in an enclosed area with lots of people around and a very good safety record. It's possibly the most perfect place to let your child cycle to their hearts content.
These three areas are the perfect place to let your child ride their kid bike. Enjoy it with them, because you know they are going to be safe.