Much of the city centre is pedestrianised and compact, so your best bet is to find somewhere suitable to park your car and set off on foot. Most of the central area is designated "Blue Zone", which means you have to pay for parking. It is currently 1 lev per hour on working days from 9 am till 7 pm, with no restriction how long you stay.
You can buy prepaid vouchers from the attendants who wander around checking the cars, or from the ticket machines on the pavement. If you have a local mobile phone account you can send a text message with your registration number to 1356.
The "Transport Burgas" website has lots of helpful information on "Blue Zone" areas and how to pay. (See link!)