However, if your local parking enforcement is done by council employees, it might be best if you completely forgot about 'yellow line violations'. Even though it's no longer a criminal offence for people to park on those lines (so the police can't take any action against people doing so), it remains a criminal offence to 'cause an obstruction' on a public highway (including blocking access to a driveway off that highway). So, once again, your local community constable might be able to assist you. (Remember to stress that your complain relates to an obstruction, not to yellow lines).
Finally, to actually answer your question:
No, you can't have private traffic wardens. Parking enforcement is a matter for either the police (or those who assist them such as traffic wardens or PCSOs) or for local councils.
Chris