Parathyroid surgery is a very safe surgery.  Complications are rare. 
 
Failure to cure primary hyperparathyroidism (to find the enlarged parathyroid gland/s) is also rare.  The success of minimally invasive parathyroid surgery in 95%+ and the success of bilateral parathyroidectomy (4gland exploration) is 92-95% in experienced hands. 
 
Post-operative bleeding is very rare.  This normally happens in the first 24hrs following surgery.  The risk is 1% or less that you may need to go to operating theatre to stop the bleeding and remove blood clot.  If bleeding happens, most/all people postoperatively require admission to Intensive Care Unit (ICU) as they require breathing tube (ETT) for 24-48 hrs.  This is due to swelling of the windpipe (trachea) and surrounding tissues. 
 
Permanent or temporary damage to recurrent laryngeal nerves is also rare.  The risk of permanent damage is up to 0.5% and temporary up to 5%.  Temporary damage should resolve within 6 months of surgery.  The outcome of damaging recurrent laryngeal nerve on one side is hoarse voice and some people develop slight shortness of breath on exercise.  Damage to both recurrent laryngeal nerves is extremely rare and may lead to problem with breathing and require tracheostomy. 
 
Permanent or temporary damage to external branch of superior laryngeal nerve is also rare.  If damaged it leads to inability to project the voice.  This is most noticeable in public speakers, teachers or people when they are trying to yell.  The risk may be up to 5%, but this damage is difficult to diagnose as there is no good test for it. This is unlikely to happen following minimally invasive parathyroid surgery. 
 
Permanent or temporary damage to parathyroid glands leading to hypocalcaemia is also rare.  Risk of permanent is less than 2% and temporary up to 10%.  For patients who have had minimally invasive parathyroid surgery the risk is 0%.  The treatment is with calcium and vitamin D replacement.
 
Infection is extremely rare as well, less than 0.5%.  It extremely rarely requires surgery and is always treated with antibiotics.
 
The best prevention of complications is that your surgeon has had subspecialist training in parathyroid surgery.  It has been shown that these surgeons have much lower rates of complications, especially damage to laryngeal nerves and parathyroid glands or to be unsuccessful in finding the enlarged parathyroid gland causing hyperparathyroidism.