I will accept an answer in the form of references to the literature about my question as well as any other information. I am quite ignorant of the area and that will be clear from my question.

I take it that interpreting a system (e.g., PA) in some structure without the use of parameters is harder than doing the same with the use of parameters. I would like to understand the difference of the consequences of the two tasks (interpreting with parameters vs. interpreting without). What does it have to say about the structure in which the interpretation takes place?