If a Muslim's parents are Muslims but they don't pray, and they don't want him to pray at mosque, but they want him to pray at home although the mosque is near home. Then what should he do?
You should educate them that if a mosque is close to your home, a man should make it a duty to go to the mosque every 5 times a day.
The Divine instruction is,
And bow your head with those who bow (in worship)
(Surah Baqarah 43)
Also some of the Prophets statements-
The Holy Prophet has also stated:
It better to join another person and pray than to pray alone and it is more superior in the company of two men and the bigger the congregation the more liked it is by Allah.
The Holy Prophet says in another tradition:
He who attended the congregation for Isha prayer, he received the reward of half the nights worship and he who attended both Isha and Fajr prayer with the congregation he received the reward of the whole nights worship.”
Praying in mosques is very important or you must have a gathering where you all pray together. Read below
The Prophet, himself never omitted even a single congregational prayer. Even in his sickness, it was strenuous for him to walk; nonetheless, with the aid of two of his companions he came (while his auspicious feet were dragging on the ground) to the mosque and prayed with the congregation.
Once the Holy Prophet was constrained to say:
I intend to tell the Muazzin (person who calls the Azaan) to call out the takbir and ask someone to lead the prayer and I myself set the house of that person on fire who did not attended the congregational prayer.
(Bukhari & Muslim)
The merciful Prophet proposes a punishment for those who do not attend the congregational prayer at the mosque, even though they perform them at home. One should ponder and be warned as regards to those who do not attend the congregational prayer at the mosque.
Also there are many hadeeths on this topic. I right now can’t recollect them. Search for them and make them read it.
If they still don’t listen leave it upto Allah as he forbades going against your parents’ wishes.
There's priority in Islam, meaning your parents are more important than praying in the mosque. It's not the same if they were trying to make you not to pray. But since praying in mosque is not Wajib and doing what your parents tell you is (as long as it's not against your religion and beliefs, even in that case it's recommended to treat them respectfully) you're good praying at home.
As for the issue, you can encourage them to pray by doing what you think is appropriate. But remember the very tick point in Quran about how to act with one's parents:
فَلا تَقُلْ لَهُمَا أُفٍّ وَلا تَنْهَرْهُمَا وَقُلْ لَهُمَا قَوْلاً كَرِيماً Surat Al-'Isrā, Ayah 23
Don't even say "of" (which is the tiniest word of complaint) to them, and treat them as respectfully as possible and talk to them kindly (even if they are angry at you).