Does seawater promote the healing of surface wounds and infections? If so, what are the likely causal mechanisms at play?

No, no, no. I've been hearing this for 30 years; swim in the ocean, your rash, wound, or whatever will heal. This is further evidence that your parents are not doctors. Seawater contains nasty bacteria, particularly in estuaries and near rocks or coral, and in areas close to fisheries, mines, farms, stormwater drains and sewage plants. People have died. If you have a skin wound or rash, stay out of the ocean!
Seawater Cure. The issue with using seawater is that it is not sterile and could add bacteria to a wound. The other part of the equation is that all wounds arecreated equal. We use sterile Normal Saline all of the time to cleanse wounds without damaging the good cells in the wound.