I do a lot of parts work with myself and I've never come across an incompatibility that the parts wouldn't be able to resolve. They often don't agree initially, but after you let them talk and express, they naturally start moving towards healing and integration. They want unity.
After hearing from both parts, I come out of them and watch them from the center consciousness and tell them what I observe about their patterns. This makes them more aware, and when I go into them again, the conversation takes a new turn. I also validate them and tell them what I think and feel about them.
I honestly never have trouble or difficulty doing parts work. Maybe it's because I never force it.
I have a lot of trouble and difficulty with CP though. Don't know why. Maybe it's because I have an aversion towards feeling and I feel unworthy of getting my inner child out of traumatic situations. Maybe I just suck at feeling and sinking into the emotion and being present with it unconditionally. It's too painful and I always just can't wait for it to end. It's mostly the initial sinking into the emotion part that I don't like and also the being stuck inside the memory. I do like and utilize other parts of the process.
It's just much much easier for me to validate and comfort my traumatized inner children in parts work than through CP. Maybe I also don't like closing my eyes. Also, when I'm doing parts work, I'm not stuck in the traumatic memory, and that feels safer. Blah blah. I don't know. Maybe I don't like just feeling the emotion without knowing why, and parts work is all about the why.
In CP, it's emotion first, and understanding comes afterwards. You have to trust and sink into the emotion. You have to jump into the unknown.
But in parts work, you are more in control. You pick an issue that you want to work with. Then when you channel the parts and the emotions come, you instantly know why. You never have to jump into the unknown. At least that's how it is for me.
Sorry, this wasn't a discussion about CP vs. parts work, but as I started writing my comment, this is what came up.