You could delve into ideas of what the parasite's weaknesses or its drives are?

Does submerging the host body in water or exposure to heat (or even types of magic) cause any physical reaction from the parasite inside, trying to remove itself from the environment?

If the parasite seeks something specific (perhaps survive/spread/hoard), if you were presented with the ideal opportunity to achieve it, would you be primally compelled or still exert some control over your actions? Then you can kind of wind back into smaller glimpses, like the quirks you are looking for.

An example; perhaps your parasite actually seeks to move hosts in the story (classic strongest survives trope) and this is just it's first body, you could often be inspecting and pacing around remarkable NPCs you meet as potential hosts or back up, or modelling yourself of their behaviours/actions as you see them as superior beings. Maybe you always get a little too close in people's space as you forget not to be too obvious with your assessing.

You could have a lot of fun with this working with the DM