These suggestions bring back so many memories from when my dc were little.
We spent many a day at Ridley Park in Blyth, dc playing in the water feature and then getting back in the car for a short drive to the beach to play in the sand, followed by fish and chips and ice cream for afters if you still had room.
Also Shiremoor wooden park, as the dc used to call it. Agree with poster who said it looked a bit run down but the dc would spend hours climbing and playing in the fort and using the little bikes which were dotted around, followed by a cheap hot dog.
Wallsend Park is worth at least a couple of hours if the weather is nice. Nothing fancy at the cafe but enough to satisfy little appetites, ice creams etc.
Saltwell Park in Gateshead. We used to start off at the play park for "little"people , then when they got bored with that move on to the larger play equipment. Then visited the rabbits and peacocks ( and talking minor bird) in the small pets area. Picnic in the rose garden where the dc could run around and watch the squirrels and then into the maze. The parents can stand above and watch the children having fun getting lost. It's great because when they have had enough of one area there are plenty of other areas to move on to.
So many hot summer days spent strawberry picking at Brockbushes near Corbridge. Nothing better than strawberry jam made with strawberries you've picked yourself.
And finally Exhibition Park , Newcastle, which I don't think anybody has mentioned. A large play area with a lovely coffee shop, adults enjoying a latte sitting under the parasols, knowing the children are safe and in sight.
Honestly I could go on and on. Wish my dc were still little. Happy happy memories.