Parker is English occupational from Old French: the keeper of a park. In medieval England a "park" was enclosed hunting ground for a lord, so the parker was a gamekeeper or park officer, not someone who simply liked green space.
The surname is a job title that went hereditary. It is unrelated to the modern given name Parker in origin, even though Americans now use Parker as a first name too. Enclosed parks were status symbols, and keeping them was skilled estate work.