I’d definitely shop around for some of that - gas and electric, phone, Sky and water all look higher than they should be and you may be able to switch provider or find a cheaper deal. Or get on a water meter, potentially (but do some research). Car insurance looks quite high - is it a valuable car? Does it include breakdown cover?
Personally, I don’t think you would have loads left. You haven’t accounted for kids’ clothes/shoes, school lunches if they have them, kids’ clubs, any wraparound care you need, car breakdown cover/service/MOT, haircuts, setting aside money each month for home maintenance, for birthdays and Christmas, or holidays….. That is a good salary but as the lone adult earner it won’t go far. So ignore your parents.
Do you receive maintenance? Child benefit?
In your position, I’d be setting up a standing order to save a fixed amount each month, possibly in more than one account to allow for different pots. Maybe one for birthdays and Christmas, and a separate one for car and home maintenance. Do it so it goes out on the first of the month and try not to touch it. If you get to the end of the month, there’s never anything left to save! I’d work out what you would like to spend or be able to spend on these things, divide it by 12, and have that as a short term savings goal. Even a small amount will really help.