Actually, it is forbidden to make fun of Islam or Muslims. Such a behavior can’t be the trait of the Muslim at all. It is also an unbecoming behavior from a non-Muslim as well.
Dr. Sano Koutoub Moustapha, professor of fiqh and its principles, International Islamic University, Malaysia, states the following: Islam always calls upon Muslims to master their religion and enhance the knowledge of the religion.

Certainly, it is forbidden to make fun of Islam or Muslims. There are many verses which condemn such behavior or attitude. Therefore, you need to remind that fellow Muslim of the graveness of the sin for making fun of Islam. Surely, this is not expected from a fellow Muslim nor it is welcome from a non-Muslim. These are some of the verses which prohibit this:
(When ye have divorced women, and they have reached their term, then retain them in kindness or release them in kindness. Retain them not to their hurt so that ye transgress (the limits). He who doeth that hath wronged his soul. Make not the revelations of Allah a laughing stock (by your behavior), but remember Allah’s grace upon you and that which He hath revealed unto you of the Scripture and of wisdom, whereby He doth exhort you. Observe your duty to Allah and know that Allah is Aware of all things.) (Al-Baqarah 2: 231)
(He hath already revealed unto you in the Scripture that, when ye hear the revelations of Allah rejected and derided, (ye) sit not with them (who disbelieve and mock) until they engage in some other conversation. Lo ! in that case (if ye stayed) ye would be like unto them. Lo! allah will gather hypocrites and disbelievers, all together, into hell) (An-Nisaa’ 4: 140)
(And if thou ask them (O Muhammad) they will say: We did but talk and jest. Say: was it at Allah and His revelations and His Messenger that ye did scoff?) (At-Tawbah 9: 65)
(And when he knoweth aught of Our revelations he maketh it a jest. For such there is a shameful doom.) (Al-Jathiyah 45: 9)