Upvote:6
Short answer is yes
According to this list from the Spanish Foreign Affairs department (which should probably be your only reference), Turkey is on the list of countries that need a Schengen visa to visit Spain. The criteria for this is very often citizenship, rather than status in a third country.
Furthermore, their list of requirements state you must have a Green Card to apply in the US, which indicates permanent residents are not exempt from a travel visa.