Named after Bay Parkway, which was earlier known as 22nd Avenue.
Its renaming as a parkway was first proposed in the state legislature in 1892, along with Bay Ridge Parkway and Fort Hamilton Parkway, to boost real estate along it. Sources differ on when the new name took hold: around 1892, says Forgotten New York; the 1930s, says Wikipedia. Bay Parkway runs about 3 miles southwest from Ocean Parkway to Bath Beach. The station was originally 22nd Avenue–Bay Parkway.