Can you have hypoparathyroidism and hypothyroidism at the same time?

Of course. The physical proximity of the two organs involved (the four parathyroid nodules actually sit on the thyroid) mean a variety of diseases and disorders can affect both at the same time. The differential diagnosis is a long one, including destructive infectious lesions, cancers, autoimmune disorders, etc.

I don't think that one would necessarily preclude the other.

Yes. In fact, if you have autoimmune hypoparathyroidism, you are more likely to have autoimmune thyroid disease (autoimmune diseases tend to cluster, although true autoimmune hypoparathyroidism is rare).

"Hypoparathyroidism" (i.e. low calcium levels) can also be caused by diabetic kidney disease. Type 1 diabetics are more likely to have more than one autoimmune glandular disease, so it would not be unheard of to have all 3.