I always look for free or cheap place to park a car and Paris is in my top worse 5 cities in Europe in this matter (hmm... the worst for me was Florence - but it's the best city in Italy).
Unique Suggestions: 1. Stay at one place and look for a car leaving a place - be first to take the place - the competition is big :-)
2. Drive around (recommended up to 1 hour), be patient
3. Park in covered/underground or other paid parkings - be ready to pay more then for your accommodation :-)
3. Park your car in a forbidden place like most do (rather not in the middle of crossword) and prepare Euros - look at my picture!
Fun Alternatives: Use subway and/or your legs/feet.
Driving in Paris is a little bit crazy, parking a car surely is crazy. I noticed that most cars in Paris are a little damaged (is it correct word?) just after small mini-crash during parking and almost all have damaged external mirrors hehe.
Unique Suggestions: Be patient hehe.
Fun Alternatives: Go on foot, use subway (metro).