The taxi driver will drop you off at the designated area in each park's parking lot. He/she will not have to pay to park as they will leave after you depart.
If you are staying at a Disney resort the Disney buses will drop you off close to each park's entrance. Also, note that no cars are allowed near the Magic Kingdom entrance. If you take a taxi to the MK you will be dropped off at the TTC and then you will need to either take the monorail or boat to the MK entrance. The easiest way to take a taxi to the MK is to have the driver drop you off at the Contemporary Resort. From there you can take the 7-10 minute walk to the MK entrance.