Professor not giving solutions to exams and assignments that I need to grade
I can't tell quite what your concern is, so I'll address both.

Is it OK for a non-professor to write solutions and design a rubric?

In my experience (large US universities, STEM), this is extremely common. I disagree a bit with others saying that only professors can make such determinations (though of course the instructor of record can make such determinations if they are inclined to do so).

Is this abusive toward the TA?

How are you being paid?


If it is hourly, then there is no issue; you should solve the problems and make a rubric however you see fit, and then bill for that time as normal.
If it is a set stipend, then there is usually a nominal time requirement (in my university it was 20 hrs/week for grad students). If making solutions will put you way over this, it's fine to raise the issue with the professor and/or with the administration (in fact, you may technically be required to do so)


One other tip -- you might consider taking the top-performing students' solutions and using them as a starting place for your own solutions. Of course, you'll have to be sure you're grading those students fairly and catching any errors. But I've found that the I can often create a solution set just by mixing and matching solutions from the top ~3 students and adding a few expository details.