What is the ruling of Islamic law regarding not performing ghusl (ritual bathing) upon waking from sleep when traces of semen are found on clothes, considering the difficulty of ghusl for the young man and his parents due to his inability to walk, and is there an alternative to ghusl?

A ritual bath (ghusl) is obligatory for whoever is certain that semen has been discharged. If performing ghusl is difficult for you or for someone assisting you, then the reward is commensurate with the hardship. Ghusl remains obligatory as long as you are able to perform it, even if by hiring someone to assist you. If you are completely unable to perform it and cannot find anyone to assist you, then perform tayammum (dry ablution). Your tayammum will be sufficient until you are able to use water. If you are uncertain about the discharge, the default is that it has not occurred. However, if you are certain of its discharge but are uncertain about its nature, then ghusl is not obligatory for you; rather, you may choose.