Emirgan Park
Emirgan is another massive park, a little farther along the Bosphorus than the other two.
It's known in particular for the beautiful flowers that are planted, and it's a must-see during the spring for the variety of colors of tulip that are planted all over in various incredible arrangements.
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.