What do you call a road that parallels a main road and is primarily used for parking?

Solution 1:

In BrE they are called:

Service road:

a relatively narrow road running parallel to a main road and providing access to houses, shops, offices, factories, etc, situated along its length.

(Collins Dictionary)

The term, apparently is used also in AmE as an alternative to frontage road as suggested by M-W:

Service road:

frontage road:

The mother’s car had stalled on a service road in Fort Worth when it was pushed by waves of water from other vehicles, the Fort Worth Star-Telegram reported. - Bradford Betz, Fox News, "Flash flood sweeps car into culvert, killing mother, 2-year-old daughter," 9 Sep. 2018

Police said protesters began to block lanes of a service road. — Fox News, "Thousands rally nationwide for illegal immigrant families separated at border," 2 Oct. 2018