Parking a vehicle fully or partially across a dropped kerb is classed as an obstruction and either the police or local council can enforce the contravention. Based on the resources a particular authority has in dealing with this, attention will usually be focussed on offences that impede those with disabilities. If you parking isn't harming anyone, blocking access to someone's property or making it hard for vulnerable road users (people pushing prams, wheelchair users etc) then I don't see how he'd get you into any trouble. But you can't block any part of a blocked kerb, even your own - legally speaking.

The public road is open to anyone as long as there aren't road marking or signs to prohibit parking (and no rules are being broken). You don't have to park on your driveway, but it does make it easier for emergency vehicles and the like to get through roads - especially if they're narrow. Similarly, someone else can park in front of your house on the public road and you can park in front of theirs legally as long as you aren't breaking any rules.