He'll be entitled to half the equity (value of house less ringfenced deposit and less any mortgage debt). Unless you're in negative equity it should hopefully equate to more than his mortgage contributions. He won't get the payments plus equity as that would be a double recovery.
He sounds ghastly considering the house is also keeping a roof over the heads of his 3 children! I expect what he is getting at is that if you're the primary parent and work PT/earn less he's put more towards the payments than you and he wants to to "pay him back" but that is tough (childcare is part of your contribution!). If it's a joint mortgage the bank won't care who has paid what.
I agree with other posters. Put a pause on the extension/renovation work. It will boost the value once complete which at the moment he's be entitled to half of even if he's not paying (like I said the bank doesn't care who pays). Also if you might need to remortgage soon part completed building works could be a spanner in the works. Wait until its sorted!
Sounds like your parents are a good support and can help you financially to a certain extent which might make all the difference in being able to buy him out if your remortgage power is limited.
Please get legal advice on what needs to happen. Likely the first (legal) step will be making him an offer on the equity in return for a transfer. If he plays silly buggers you might have to apply to court for an order for sale (transfer to you effectively). If you get this and he still won't sign the judge can sign on his behalf.
As part of that process the court will need to be satisfied that you have the finances available in one form or another (including remortgage offer) so get working on that. Tell your mortgage provider that you have split, that you'll be paying the mortgage on your own from now and enquire about remortgage upon buy out and transfer. You might need to see a mortgage broker to look at different types of mortgage/providers.
Try and grey rock his aggression and call the police if you are worried (every time).
If he's not paying maintenance put in a CMS claim pronto. You will need that income to keep the family afloat.
Good luck. (Disclaimer: I'm a solicitor but don't practice family law.)