If you live in the village of Ardsley there is no permit parking. You can have an Ardsley address, but be in the Town of Greenburgh, that would get you a permit at the Hartsdale Station. You can buy parking in Scarsdale and in White Plains. You can also try Irvington and Hasting. Dobbs Ferry has a lottery of 25 spots for Ardsley residents.
The best place to check regarding parking permits for the Scarsdale Hartsdale and Dobbs Ferry Metro North stations is to go to the MTA website. You can access each individual train station and within each page your will see information regarding parking with contact information.
Here is the parking info for the Scarsdale Station: