Is it permissible to use money from a father's salary or end-of-service gratuity, after he has become unable to work and was receiving a salary for doing nothing, and to use this money for study expenses?

If the officials are authorized by the employer to transfer the father to a school almost devoid of students, and in consideration of his health condition, then there is no sin upon him in taking his salary. The basic principle is that there is no sin upon the questioner in eating from his father's wealth and benefiting from it, because the default assumption for what is in a Muslim's possession is that it is his property. If we assume that the father is not entitled to the salary for the days he did not attend, then he is entitled to the salary for the days he did attend, and thus his money would be mixed. Dealing with someone whose money is mixed is not forbidden. The end-of-service bonus is tied to the full duration of service, and some of it is free from any issue.