We’ve been to the playground in Verulamium Park in St. Albans a few times – it’s only 20 mins by car and the car park is free for a couple of hours on a Sunday. The park itself is large, and features Roman remains dotted here and there.
There’s also a paddling pool with a cafe next to it. The one time we took our little one’s swimming gear he didn’t want to paddle and it was very busy, but it’s a great feature.
It’s a short (but quite steep) walk up from one end of the lake into town. This way you pass the rear of the cathedral, whose grounds are also a great place for (supervised) kids to play.