Who you should contact depends upon where you live.
In major conurbations, the routine regulation of parking is undertaken by wardens working for the local council. Outside of these areas, it's still the police and traffic wardens that do this job. (The police can still 'issue tickets' within major conurbations but they no longer regard this as part of their day-to-day duties).
I live in a small town where the two designated local police officers spend a great deal of their time knocking on people's doors to advise them that their vehicle is illegally parked. (They also mention the issue frequently in the local newsletter). Residents who repeatedly contravene the rules 'receive a ticket' from these officers. If you live in a similar area, your 'Community Constable' is the person to speak to.
However, I'm not far away from a large town (Ipswich), where parking matters are dealt with by council wardens. Anyone living there, with problems similar to your own, would need to contact the council. ('Community constables' might help in some circumstances but, generally, all they would do would be to refer the matter to the council's parking service).
If you see uniformed parking wardens, employed by the local council, patrolling 'yellow-lined' areas near to you, contact the council. Otherwise, contact your local police officer.
This might be of interest:
http://www.theanswerbank.co.uk/Law/Question340 152.html
Chris