I've been getting muscle cramps and spasms a lot lately. Is this possibly a thyroid or parathyroid problem?

Not the most likely- Causes. Thyroid problem could possibly cause cramping but other symptoms are far more likely to be seen first in your thyroid isn't working right. Since one of the functions of the parathyroid is to control calcium levels, and low calcium levels can produce muscle spasm, it is also possible. Nevertheless, thyroid and parathyroid problems shouldn't come at the top of the list of considerations.
Medical Evaluation. Hard to say offhand because many things can cause muscle cramping. Please discuss with your doctor. You may require a work-up to yield more information for a diagnosis.