Gotta go with Reid, since that's my DS's name. The Reid spelling is the Scottish variant, and the Reed spelling is, I believe, the English variant. So they are both legit spellings, and if you have ancestry in either camp that you'd like to honor, this may help you make a decision (my ancestry is Scottish, which is why we chose Reid). Otherwise, it's just personal preference. This board tends to strongly favor the Reid spelling for whatever reason. And I think, though it's been awhile since I looked, Reid is the more popular spelling nationwide.
I actually do like Parker, too. But it falls a little into the trendy category for me due to the -er ending. The "do-er"/profession names are popular right now (e.g., Archer, Hunter, Carter). But it's not so trendy sounding that people would hear it and automatically think of this decade, so this may not be an issue. Good luck deciding!