There is a market-clearing price. Call it $N. That's the level at which people willing to pay $N can always get a parking spot. For a smaller value of $N, someone wants to get a parking spot by paying $N can't. (That price might even be $0 if you have more supply than demand. Some economists will talk about "The High Price Of Free Parking," and there's something to that, but if the city decides to make the market-clearing price $0, then they should do it by always having enough supply available.)

If you decide to distribute the limited resource in a fashion besides money, such as a lottery, you have an inefficient system. Instead of paying money, people have to drive in circles, or queue, or wait to find out the results of a lottery. No one benefits by this. If you are trying to help the poor, just give them more money, and they will decide on their own how valuable parking is to them.

For the record, I fully support the city deciding that these other people don't get to extract the city's rents. The city should stop people from occupying a public space merely because they want to get paid to move on. It's the city's resource to sell. (And maybe the market clearing price becomes $0 once you kick out the people who are making a problem just to be paid to stop making it -- which is the definition of a racket.)