Does osha allow a construction worker wear shorts for work?

Whether or not shorts should be worn is not a matter of OSHA requirements. It is a matter of risk in that environment. If an employee's exposed skin is at risk because of the operation he is doing, then he needs to cover it up.

I would venture to say that there are many jobs within manufacturing that could be done safely wearing shorts, and many that are too high risk.

What OSHA expects employees to wear is a function of employee exposure not fashion.