Getting to Pärnu is a breeze, but finding a convenient place to park your car can be a challenge in summer. In the downtown area, free parking is available in the lot on Kalda street near the bridge and on all of Tammsaare pst. Most other streets in the downtown area are marked with blue ‘P’ signs informing you that they are paid parking zones Mon - Fri 08:00 - 18:00. Here parking is free for those staying 1 hour (near the beach 30 minutes). Others must pay €1/hr or €4/day. Tickets are sold by vending machines on the street (coins required), and at R-Kiosks like the ones in the bus terminal and at Rüütli 32. Whether or not you’re paying, you must leave a time marker or a note on your dashboard indicating the time you parked. From June through August 10:00 - 18:00, parking on streets near the beach also requires payment. Here a 1-hour ticket is €5 and a day ticket is €12. 
For more info see http://parnu.ee/index.php/en/home/transportation-and-parking