I strongly recommend publishing and adopting the following policies:
Publish detailed grading rubrics describing how you assign partial credit. Follow them slavishly when handling regrade requests.
Regrade requests must be submitted in writing at most X days after the graded work has been returned. Each request must include a brief written explanation of why the grade is incorrect. (Note: not "...why they deserve more points.")
While you are happy to discuss answers and even grades in person, you will not change any grades in the students' presence. Or maybe even within 24 hours of meeting with the student requesting the regrade. Maybe add an exemption for arithmetic / recording errors.
Your responses to regrade requests are final. Further appeals will be automatically forwarded to the instructor. (If the instructor bounces them back to you, that means they're happy with your earlier decision.)
Finally, ask the instructor to announce these policies on the course web page. Better yet, ask the instructor to forbid you to change grades in the students' presence, or to regrade the same submission more than once.
As for demands from students for higher grades: Look them straight in the eye and tell them that giving them anything less than the fair and honest evaluation that they (or their parents) have paid for would be grossly unfair, not only to other students or future employers, but to the students themselves. Refer all complaints about your grading rubric to the instructor. Unless they want to discuss the substance of their work, gently but firmly kick them out of your office.