For myself with limited walking ability The Rec off Hiltingbury Road in Chandlers Ford is perfect, with ample parking, and the circular walk is just enough for me. However during term time it is more difficult to park. There are many varied events which are held at The Hilt. The dog walkers are friendly and welcoming and the dogs are pretty good too.
Valley Park in Chandlers Ford is also a very nice walk with a pleasant stream which most dogs love to splash in. A longer walk. Parking available.
Farley Mount is a much larger area. Lovely.
Fleming Park old golf course