His parents think that he does not offer the naafil prayers, but he prays them in secret for fear of showing off. Should he tell them about that?

Praise be to Allah.

Firstly: 

Concealing acts of worship and not doing them openly is better than doing them openly and announcing them. Allah, may He be exalted, says (interpretation of the meaning):

“If you disclose your Sadaqat (alms-giving), it is well, but if you conceal it, and give it to the poor, that is better for you. (Allah) will forgive you some of your sins. And Allah is Well-Acquainted with what you do”

[al-Baqarah 2:271]. 

Ibn Katheer (may Allah have mercy on him) said: The words “but if you conceal it, and give it to the poor, that is better for you” indicate that giving charity in secret is better than giving it openly, because that is further removed from showing off, unless giving it openly will serve a greater purpose by setting an example for people, in which case it is better in that respect.

End quote from Tafseer al-Qur’an al-‘Azeem, 1/701 

Secondly: 

The Sunnah indicates that it is better to offer the naafil prayers at home. Al-Bukhaari (6113) and Muslim (781) narrated that the Prophet (blessings and peace of Allah be upon him) said: “You should pray in your houses, for the best of a man’s prayer is in his house, apart from the obligatory prayers.”

The scholars (may Allah have mercy on them) stated that one of the benefits of that is that praying at home is closer to sincerity and further removed from showing off. 

Ibn Qudaamah (may Allah have mercy on him) said: Offering voluntary prayers at home is better, because the Messenger of Allah (blessings and peace of Allah be upon him) said: “You should pray in your houses, for the best of a man’s prayer is in his house, apart from the obligatory prayers.” Narrated by Muslim. It was narrated from Zayd ibn Thaabit (may Allah be pleased with him) that the Prophet (blessings and peace of Allah be upon him) said: “A man’s prayer in his house is better than his prayer in this mosque of mine, apart from the obligatory prayers.” Narrated by Abu Dawood. And that is because praying at home is closer to sincerity and further removed from showing off.

End quote from al-Mughni, 1/443 

For more information, please see the answer to question no. 22209 

Thirdly: 

If not telling your parents will result in what you mentioned, then in this case it is better to tell them, so as to ward off suspicion and negative thinking of you. Moreover, perhaps by telling them – as you said – it will make them feel happy, and making parents feel happy, by obeying them and doing good, is part of honouring them, which is something that is prescribed and that one should be eager to do. 

With regard to the fear of showing off, you have to strive to make yourself sincere, and ask Allah, may He be glorified and exalted, to help you to do that. 

May Allah help us and you to be sincere in worshipping Him, may He be exalted, and may He protect us all from the evil of our own selves, for He is able to do that. 

And Allah knows best.