In general the risks are significantly less than those of thyroid surgery.
There is a very small chance ( less than one in 500 patients) of bleeding requiring a reoperation. The risk is increased if you must take a blood thinner.
There is a very small chance ( less than 1%) of temporary weak and husky voice due to vocal cord palsy. This can happen if the nerve to the vocal cord is traumatised ( stretched, bruised) in the process of cutting the parathyroid adenoma away from it.
Permanent voice damage due to vocal cord palsy is even less common.
Neck infection is rare.
Unsightly scar is possible, but in general the skin suture technique is meticulous and results in a well healed and very fine scar.
Hypoparathyroidism ( under active parathyroid function) is only possible if all four glands are removed or damaged in the course of surgery. When all four glands are abnormal and 3 or more glands need to be removed to correct the problem, there is a small chance of hypoparathyroidism.
Depending upon the surgical strategy, the chance of having persistent hyperparathyroidism is from 2-5%. The chance of recurrent hyperparathyroidism increases with time and is at least 5-10% over many years as new adenomas can grow on previously normal glands.
General risks from surgery
For fit and healthy patients there is the rare possibility of reaction to an anaesthetic drug, antibiotic, or other agent. Blood clots: Deep vein thrombosis or Pulmonary embolism are extremely unlikely with this kind of surgery.
