Is it obligatory to obey one's parents regarding refraining from applying kohl on Friday, or to prioritize the application of the Sunnah in this matter?

Applying kohl on Friday is not from the Sunnah, as the Prophet (peace be upon him) did not do so specifically. Rather, he applied kohl and encouraged its use for the benefit of the eye, without restricting it to a specific day. A man's use of kohl for beautification is generally not acceptable, and Ibn Uthaymeen paused on its permissibility. Doing so may lead to people having ill suspicions about the person, and a Muslim should not put himself in a position of doubt. If the parents dislike it, then obedience to them is due to their concern for their child.