Short answer: No, they are not synonymous.
If you think you are partly to blame, then you think or suspect that there is enough blame for everyone involved, perhaps including Peter and Andy, and that others may share the blame with you.
Depending on what share you think you or they deserve, you might say:
- A small amount; less than a majority of the blame/fault
- A large amount; a majority of the blame/fault
- The complete amount of blame; all the blame/fault
If you say a part of you thinks you are to blame, it indicates that you are not certain that you are to blame, either partially or entirely. You are of two minds: you are undecided, or have two different opinions and can't choose between them:
- I was at least partly to blame for the argument; I did something that caused it.
- I was not to blame for the argument; whatever I did, it was nothing to cause it, or someone else did something that caused it.