Most poor people don't own their house and often don't have a car – if you do own both you're middle class.
I'm of the opinion that when public goods are cheap enough to face shortages all the time the market economy steps up because better off people will spend more to save time/hassle.
The problem is when things are expensive enough to kick out a lot of people, but not enough people actually alleviate shortage, which is basically how it currently goes with parking.
No, it's the opposite. A city built around everyone having a car makes car ownership a cliff. Normalising not having a car (and a reliable bus service - like the kind you get by turning street parking spaces into bus lanes - helps with that) makes the ladder gentler. If people are late for work because they couldn't find a parking spot just as often as people are late because the bus was late or didn't show, maybe there will be fewer horror stories of people getting fired because their car broke and they couldn't afford to get it fixed.
If you're worried about the transition, subsidize other forms of transport and build that out first, but forcing poor people to own cars just to make it to work is not a good way to help them.
Sorry for the cynicism, I'm actually for increasing the price of parking, but recent political events have robbed me of any illusions that environmentally friendly policies have a future. When they have a choice between the environment and paying less money (short-term), most people will choose paying less money.
Outside of maybe a couple urban areas like NYC, that's patently untrue in the United States. It would not surprise me if the poor frankly had more and older/rougher cars than their more wealthy counterparts.