Can my HOA restrict the areas where my children can play? For example, can my HOA restrict playing to a designated playground?

No.  It is illegal for an HOA to set up a playground, and then to declare that children can only play on that playground and nowhere else in the complex.  Children are free to play in open lawns, sidewalks, and open areas.  The HOA cannot restrict this activity. Any rule or CC&R that declares that children can only play in a designated area would not be enforceable.