Is there any reason to keep lower-tier ships equipped?
I just reached level 30 (unlocking my Tier 4 ship), and noticed that some of my Tier 3 ship's equipment is better than the defaults, and that I'd yet to encounter a reason to go back to any of my older ships.
Is there any reason I would need to use my old Heavy Cruiser, or am I safe stripping it for parts?
You're safe to strip older-tier vessels for parts. The main reason you wouldn't is if you were playing some sort of hybrid class that liked to swap between two vessel types (escorts and cruisers) depending on if you were doing PvP or PvE combat. Even that case is pretty much nullified the moment you can afford to purchase a second ship of the same tier as your main ship (although I would avoid that until you hit the max tier).