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It is apparent that chlorine has become an occupational hazard for your son. If he is allergic to cholrine, then maybe anti allergics or inhalers used for asthma may help. However, true allergy will only be eliminated on stopping the use of the pool. Alternatively your son can try for a job at pools which do not use chlorine for purification. Lifeguard jobs at natural water resources like sea beach is also an option.
It is difficult to comment beyond this at this stage. Maybe you can seek another opinion. Please let me know if there is any thing else and do keep me posted. Take care!
Im a lifeguard at an indoor waterpark as well. I have recently developed this same cough. It keeps me up at night and only seems to be manageable when I'm actually at work. A few of my friends who have been lifeguarding for several years told me that after so long, your lungs get soused to the clorhine that breathing in an environment without it is hard. The cough usually goes away in the summer when the outside air is more warm and humid, it's more prominent in the winter when the air is cold and dry. It'll go away after about 6 weeks of being away from a clorhine environment, but it can flare up again if exposed. In the meantime, cough syrup can help at night, as well as inhalers and a humidifier for the bedroom. Make sure your boys rinse off before coming home, and shower as soon as they get home to prevent other side effects of clorhine such as swimmers xeriosis and athletes foot/jock itch. Most doctors will prescribe an inhaler and skin cream to lifeguards to help keep them healthy. This cough can open them up to other things such as lung infections, pneumonia, and bronchitis as long as traveling chest colds. Make sure they get plenty of vitamins C and D until it clears up, and for several weeks after. It's painful, it makes my lungs burn, but if you take these steps, it should be a little more comfortable for them. Hope this helped.
I now live where there is chlorine in the water and i cough after any use or from in the hot shower and have a tightness in my lungs so I drink a small glass of wine and it stops dead in its tracks. I now only drink spring water or artisian water, No problem.