Praise be to Allaah.
It is not Sunnah to put on kohl on Fridays. It was not the
practice of the Prophet (blessings and peace of Allah be upon him) to put
on kohl on this day in particular. Rather what is narrated in the Sunnah
is that the Prophet (blessings and peace of Allah be upon him) put on kohl
and encouraged others to put on kohl because it is beneficial for the
eyes, without restricting it to any particular day, whether Friday or any
other day.
See the answer to question no.
132208.
For a man to put on kohl as an adornment is something that is
not acceptable in most societies. Shaykh Ibn ‘Uthaymeen (may Allah have
mercy on him) was uncertain as to whether it is permissible; see that in the
answer referred to above.
Some negative consequences may result from doing it; people
may think badly of the one who does that and some people may find it weird.
It is not appropriate for a Muslim to put himself in a position where he
will be thought badly of.
If we add to that your parents’ objection to this action,
then you should obey them, because they are doing that out of love for their
child and the wish to keep anything that may adversely affect him away from
him.
And Allah knows best.