Can one have hypothyroid and hyperparathyroidism? what does both at same time mean?

Yes: The thyroid and parathryroids are 2 separate glands and have separate functions. There are some very rare conditions where you can have hypothyroidism and hyperparathryroidism, but most likely these are two separate things. If you are having these conditions I would suggest you see an endocrinologist so further studies can be done.

There may be?: Coexistence of Hypothyroidism and Hyperparathyroidism has been studied and there is some evidence that prolonged TSH stimulation which happens in Hypothyroidism may lead to Hyperparathyroidism in RAT MODEL as the study was done on the rats only Evidence is only derived from the study on rats.There has not been any study in humans This remains a hypothesis

Yes: Yes you can have both conditions at the same time. They are two separate conditions but common so they could occur in the same patient.