As others have said, (1) party affiliation is an oversimplification of political beliefs but (2) many many people do broadly hew to one or another party line. But precisely because #2 is true, you can get much the same information as you do from "Democrat or Republican?" by asking one or two questions addressed more directly to the issues.

At least once in the past (probably much more often) researchers have done a political survey, done a principal-component analysis (or something similar) on the results, and published their conclusions about the two-ish main collections of issues on which people in their sample divide. You might want to look up at least one such case (unfortunately I remember no details, and I have no reason to believe that my google-fu is stronger than yours) and use that to decide what actual question(s) to ask.