Is the parents' condition that the divorced young man pay the alimony for his ex-wife and son before they approve his proposal to her considered an arbitrary or logical condition?

It is not a condition for a woman to marry a man that he settle any outstanding maintenance payments to his wife or child. It is not possible to definitively state whether the family will approve or not, as they may be aware of the man's financial situation, which could threaten the stability of married life. The woman can try to persuade her guardian to agree to the marriage and seek mediation from righteous people. If she is unable to do so, and she strongly believes that the guardian is preventing her marriage without a legitimate Shar'i reason, she may refer the matter to the Shari'ah court. The mere fact that a man is divorced is not a Shar'i impediment to his marriage.