To be honest, I didn't think Italy was exactly full of things for children, partly because they don't consider children to be odd things which need special treatment. However, you'll have a great time and there's a few things around - will you have a car? Tirrenia is between Pisa and Livorno isn't it?
There's a theme park at Donoratico south of Livorno. Didn't go but the leaflet looked ok. And there are a few aqua parks around - one near Cecina, again the other side of Livorno but should only take 45 minutes or so by car.
I started a thread some time ago about things to do in Florence and some kind person there suggested Green Park near Pisa. Again we didn't go but it sounded good. I'll pull out the link if you want.
And another kind person (Califrau I think) recommended going to Lucca and hiring a bike to cycle round the (wide) walls. This was really great and they do them with trailers on the back for smaller children. We all really enjoyed that.
Parco di Pinocchio was crap though. I wouldn't bother tbh. I'd go to Montecatini Terme, ride on the merry go round then get the funicular up the mountain to Montecatini Alto, where there's not much, but there's a really nice square to have lunch.
And the food's fab. I'm very jealous.