Assalaamu alaykum wa rahmatullaahi wa barakaatuhu. We have only one umbrella, so when it is raning, we have to make a difficult choice: either my father can go, or I can. We cannot go both. So who should go to the masjid? How severe does the rain have to be for us to not go to the masjid? We fear that we may catch a cold as your clothes can get wet, if we do not use an umbrella.

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger. 

The owner of the umbrella is more entitled to use it. If it belongs to your father, then he is more entitled to use it. If it belongs to you, then you are more entitled to use it. If you choose to let your father use it so that he may go to the masjid, then this is considered favoring another person with a good deed that draws him closer to Allah because he would be using the umbrella to attend the congregational prayer. The Hanbali scholar Ibn Rajab wrote, “The apparent indication of Imaam Ahmad's statement is that it is disliked to favor another person with a good deed that draws one closer to Allah, even if it is done by the son in favor of his own father. Another view reported on his authority is that it is permissible.”

You should know that heavy rain that causes harm to people is a valid excuse for Muslim men to miss the congregational prayer at the masjid. ‘Abdullah ibn ‘Abbaas, may Allah be pleased with him, said to his mu'aththin on a rainy day, “After you say, ‘I testify that Muhammad is the Messenger of Allah,’ do not say, ‘Come to the prayer,’ but say, ‘Pray at your homes.’” [Al-Bukhaari and Muslim]

As for the criterion of the heavy rain warranting missing a congregational prayer, scholars of Fiqh stated the permissibility of missing the Jumu'ah (Friday) prayer and the congregational prayer in rain that wets the clothes, and other scholars described it as the rain that requires covering one's head.

Ibn Qudaamah wrote about the valid excuses warranting missing the Friday prayer and congregational prayers, “Missing them is excused due to heavy rain that wets the clothes and mud that harms a person in his body and clothes...” [Al-Mughni]

The Maaliki book Ash-Sharh Al-Kabeer reads, “The valid excuse for missing it (Friday prayer) and for missing congregational prayer is intense mud ... that would make common people take off their shoes, and intense rain that makes them cover their heads...”

The Shaafi‘i book Al-Ghurar Al-Bahiyyah Sharh Al-Bahjah Al-Wardiyyah says about the valid excuses for not attending the congregational prayer, “...and heavy rain that wets the clothes during the night or the day...”

Allah knows best.