How can teaching assistants deal with grade disputes?
In the United States, the professor responsible for the course should take responsibility for resolving disputes (but not requests for corrections and clarifications).  There is an expectation that they will be heavily involved in grading, even if the bulk of it is done by teaching assistants.
In some cases, the professor may choose not to involve themselves in grading at all, and leave it up to teaching assistants to resolve disputes.  I do not agree with this practice, but if you are stuck with it, you can handle disputes by:

Using a rubric to grade.
Resolving all disputes using the rubric.
Refusing to compare the work of different students.  In the USA, comparing students' papers runs a risk of violating privacy laws.

Ultimately a portion of students will never agree with you about how you grade (they want to learn something different from what you are expected to teach) and this simply must be tolerated.