I'd like to create a secondary set of armor to use while soloing that improves my DPS without requiring me to switch specs. Some of the protection spells require dodge or parry to proc, and I know dodge / parry also increases my critical strike chance due to Riposte, but I'm still curious if I could improve my DPS by replacing dodge or parry with Critical Strike or Haste anyway?
A TL;DR answer is YES.
Actually a tank specification with dps gear set is still valuable when leveling - it makes you not so easy to die, gives you the ability to pull multiple mobs to make questing more effective.
However at max level, I strongly do not recommend you do so because it will make you weaker as a tank in dungeons or raids. As to solo some old raid... I have to say warrior is really not a good class for soloing because of lack of healing spells. If you like solo, hunter or death knight might be good choices.