There are squirrels running around and many different playgrounds (including some excellent climbing jungle gyms for older children).
Dolmabahçe Palace
The palace gardens at Dolmabahçe are free of charge and impeccably well maintained.
They are quite large and along the waterfront there's a café which serves snacks and drinks with the best view you'll find anywhere in Istanbul right along the waterfront.
Kids are encouraged to run around and enjoy the beautiful gardens, and in the summer months there are lovely flower arrangements to admire as well.
Atatürk Arboretum
The Atatürk Arboretum in Istanbul's Bahçeköy area is one of the most well-known places where visitors enjoy fresh air while learning a bit about botany.