The biggest one in Brighton East is Dendy Park which accommodates all age levels. It has a great playground for very young children, and is popular with athletes for jogging and workouts.
I enjoy taking my grandchildren for a walk to the Lucas St park just off Hampton St. It has a good playground that suits both very young and ages to about 7. I particularly like it because there is very little traffic, lots of big trees, and plenty of room to kick a ball. Very quiet and tranquil. Best advice is to just look on Google maps.
Another couple of favourites not too far away but will need to be by car is firstly just over the Nepean, in Jasper Road. Called Halley Park, it has a great playground for all ages, and an outdoor fitness centre. Plenty of BBQ facilities and big shade trees on the perimeter. The other one is the Thomas St South Reserve in Hampton. An amazing adventure style structure for any age and quite unique. It also has a large field next to it for ball or athletic activity. Both of these have toilet facilities.