To stop salmon patties from falling apart, don't skip the egg, mayo, and breadcrumbs. You’ll want to make sure that all of the ingredients are thoroughly mixed and chilled in the refrigerator for at least 30 minutes before molding them into patties. The salmon patties shouldn't be too dry or too wet either. If it's too wet, add more breadcrumb, and if it's too dry, add more mayo. In addition, finely chop the fresh parsley and never swap it with the dried version.