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 so that wheelchair users can safely get onto and off of the pavement rather than for driveway issues. Complaints can be made to the local police via the non-emergency 101 number, although it's usually better to contact the local council first (unfortunately they just seem to be passing the buck in your case). If the problem persists, the council can mark a white line below the dropped curb. However, the markings aren't enforceable by police or councils - they're only an advisory area as where not to park.

I'd try having a read of this (www.honestjohn.co.uk/news/legal-motoring-advice/20.../). But, annoyingly, it can be a lot of hassle to get things sorted when it comes to parking trouble because the resources aren't there to handle every complaint the council receives.