I've also read about parathyroid imbalance and 'psych' symptoms, both in hyper and hypo- parathyroidism.

Psych symptoms tend to present in all physiological imbalanced states, and it would be an unusual leap to go to the parathyroids first, unless there were other symptoms accompanying associated with high or low calcium and phosphorus.

Taking a glandular would only help if a person was Hypo-P, as hyper-P + glandular would be like throwing oil onto a fire..IMO. Glandulars are tissue elements of the gland thus still able to release hormones of that gland (why some surgeons have stitched a part of a parathyroid gland into people's arms to try to save function when doing a parathyroidectomy).
In this case the hormone would be PTH, and in a hyper-P state there is already too much PTH, causing a further increase of calcium levels and hypercalcemic symptoms.

Calcium, phosphorus and PTH blood panel would be the best way to determine whether there is a parathyroid issue, and whether hypo or hyper.
I personally wouldn't just 'try out' a glandular when unsure what state of function that gland is at.

YES, YES and YES.

Low thyroid is a common cause of depression.

I found great lift in spirits after taking NOW Thyroid Energy capsules and Standard Process Thytrophin PMG glandular. Both are available on Amazon.

There are some most common symptom of Parathyroid are Loss of energy, don't feel well, don't quite feel normal. Feel old and always in Depression etc. If you notice these type of symptoms, then you can meet with any professional psychologist.