"Dinosaur!" she yells as we discover the apple-green cartoon raptor ride. And, best of all, "Choo-choo!" at the little trains chug-a-chugging around a circuit.

There are rides and slides and playparks, all toddler-sized and decorated in bright primary colours and Peppa's signature sky-blue and piggy-pink. There is Miss Rabbit's Helicopter Flight, which gently swoops us through the air as little hands point out everywhere we can see Peppa, below us "There! And there, and there, and there…"

There is Grandpa Pig's Boat Trip, with squat, red boats bobbing around in a circle. The gloriously surreal Dinosaur Adventure lets us straddle George's favourite toy dinosaur, much as you might a horse, and zoom across a lawn, but best of all is Daddy Pig's Car Ride, with steering wheels in the front, letting my daughter "drive" me around. She takes her responsibility very seriously and demands "more" as soon as the ride stops.

"More" is what most toddlers demand of Peppa Pig, which is now in its third series (the next will be her last). It is phenomenally successful, screening in 188 countries worldwide, and is now the biggest pre-school franchise in Britain. Peppa toys were Britain's bestsellers for under-fours in 2010, overtaking Thomas the Tank Engine for the first time. Merchandising sales reached £200 million last year.

And yet it feels worlds away from the corporate world of other large-scale children's brands. "Peppa feels like a daughter to us," explains Phil Davies, one of the show's creators, as we sit at a sunny table in a corner of the park. Astley Baker Davies, the trio of animators and producers behind Peppa, have been drawing her and her family since 2002.

"She feels exactly like one of our children," agrees Neville Astley. "The show is a whole family experience," he adds, as much of the humour is about Peppa's parents and grandparents.

The family theme is important. "Paultons is a family business, which impressed us from the off," Davies says. "We were stunned that anyone was interested in doing this – and investing so much."

The Mancey family, who opened Paultons in the early 1980s, came to them with the idea two years ago, and it has cost £6 million to develop.

Paultons feels very different from corporate-owned parks such as Chessington or Alton Towers. The main park, on the original site of Paultons House, has rolling lawns, exotic bird aviaries and forest walks, as well as more mainstream rides – aimed at slightly older children – giving it almost the feel of a country retreat rather than a high-octane theme park.

The family-friendly attention to detail is impressive. All of the lavatories have additional toddler seats, for example. There are low sinks for little hands, and Daddy Pig's Big Tummy Café sells baby yogurts and tiny sandwiches, and has rocket-shaped high chairs (no hot meals, though, which is a shame).

Everything is small, from the slides and climbing frames to the boats and trains. In the playpark, everything is 3ft high, so parents don't have to shadow their children nervously.

There are slightly surreal touches that appeal to adults, too. The piped safety instructions as you join the (small) queues for the rides are in the same voice as the cartoon's voice-over. The ducks on the pond are 3D versions of the ones in the show, and circle around the water, quacking.

If the weather turns, there is a large indoor play area with padded climbing frames for different ages and a brilliant ballpark with air tubes that suck up the balls and shoot them through the air. There are little houses to walk through, including Peppa's home, complete with talking human-sized animatronic members of Peppa's family, making pancakes.

The reason for Peppa's enormous profits becomes clearer in the gift shop, where we pop in, exhausted, on the way out. Row upon row of stuffed Peppa toys stare out next to shelves of Peppa cars, Peppa clothes, Peppa tricycles – even a Peppa Pig bed. "It's a bit overwhelming, seeing it all together like this," says Davies, who has wandered in behind us.

Roma totters over, hugging a pair of Peppa Pig wellington boots, and almost swoons with joy when I buy them. Maybe there's hope for those muddy puddles after all.

• Paultons Park is on the edge of the New Forest, just off exit 2 of the M27, or a15-minute taxi ride from Southampton railway station. £19.50 (adults and children); those under one go free. Family of four: £74. www.peppapigworld.com.

More theme parks for young children

Gulliver's

There are four family-run Gulliver's Parks, including two by Milton Keynes – Gulliver's Land and the newest addition, Gulliver's Dinosaur and Farm Park. Reports suggest the parks need updating, but all have rides for two-year-olds and up, and there's an indoor Splash Zone water park in Milton Keynes, making it a good choice in bad weather.

£14.75 (adults and children); toddlers under 90cm go free. www.gulliversfun.co.uk

Thomas Land, Staffordshire

Part of Drayton Manor Park, Thomas Land has 12 Thomas the Tank Engine-themed rides, as well as play areas and a shop selling Thomas & Friends toys.

£2.50 (ages two to three), £20.95 (four to 11), £22.95 (adults); www.thomaslanduk.co.uk

Legoland, Windsor

Celebrating its 15th birthday this year, Legoland is aimed at young children (three and up), with 55 rides, live shows and a driving school. New this year is the Atlantis underwater adventure. Duploland is aimed at under-fours and has gentler rides, a playpark and a water play area.

£37.20 (adults); £26.60 (children), under-threes go free; www.legoland.co.uk

Diggerland

Four outposts of Diggerland (in Devon, Kent, Yorkshire and Durham) offer 18 rides themed around full-sized construction machinery, as well as allowing children to get behind the wheel of real machines such as JCBs. There are height restrictions: those under 90cm cannot go on any rides.

£14.99 (adults and children); www.diggerland.com

• Prices quoted are for online bookings; tickets will cost more if bought on arrival.