Price parking at the market rate. Demand for other forms of transportation increase substantially. Provide it. Poor people can now take cheap buses and trains instead of expensive cars.
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.
Price parking at the market rate. Political competitor criticizes you for being an "eco-dictator" and promises a return to free parking. You lose the next election.
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.
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.