Parents are hardwired to protect their children. They will go to elaborate steps to put themselves between their kids and a perceived threat. Some parents spend a huge chunk of their time worried about every possible thing that can go wrong, the result being that they overprotect their child from the simple accidents and incidents that help them learn and grow. Barring a child from using playground equipment is going too far. It is much better to set reasonable rules and take steps to ensure that the equipment is as safe as possible.
If the family has their own swing set, then parents need to be diligent about keeping all the swing set parts in working order. They should schedule regular inspections to check for rust, cracks, and other indications that parts are wearing out, broken, or need to be repaired or even replaced.
A few simple rules are sufficient to keep kids safe in their play. Younger children should not be allowed to use the swing set without adult supervision. Older kids should be cautioned about specific use, such as not standing on the swings or tampering with the equipment. Swing sets should be off limits during bad weather or after dark (unless there is special lighting available). Sensible rules that are consistently enforced will go a long way towards keeping kids safe on a swing set.