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According to [the USDA](https://ask.usda.gov/s/article/Is-it-safe-to-refreeze-food-that-has-thawed), your parents' method is safe:

 > If raw or cooked food is thawed in the refrigerator, it is safe to refreeze it without cooking or heating, although there may be a loss of quality due to the moisture lost through thawing. After cooking raw foods that were previously frozen, it is safe to freeze the cooked foods. And if previously cooked foods are thawed in the refrigerator, you may refreeze the unused portion.  Do not refreeze any foods left outside the refrigerator longer than 2 hours. If you purchase previously frozen meat, poultry or fish at a retail store, you can refreeze if it has been handled properly.


As for food quality, unless the food is snap-frozen, water molecules will expand during the freezing process making the food 'mushier' after repeated thaw/freeze cycles.  If they were hoping to make a seafood salad, the dish probably wouldn't look that appetising, if however lobster mornay was on the menu, they probably wouldn't notice much difference.

Quality sources:  [Today - Yes, you can refreeze thawed food](https://www.msn.com/en-us/health/nutrition/yes-you-can-refreeze-thawed-food-%E2%80%94-and-9-other-freezer-myths-busted/ar-BB19P81d) - See #3  
[bon appetit - How Many Times Can You Actually Thaw and Refreeze Food?](https://www.bonappetit.com/story/when-can-you-refreeze-food) - third para