Does verse 59 of Surat An-Nur require permission to be sought before entering upon parents if the door is open and the curtains are raised, and is it permissible for two brothers to share the same bed in one bedroom? If it is not permissible, then what is the solution?

The Shari'ah has made it obligatory upon parents to raise their children to seek permission before entering upon them during times of 'awrah (intimate times) (before Fajr, at noon, and after 'Isha') to cover their nakedness and preserve the sanctity of the parents. Once a child reaches puberty, it becomes obligatory for them to seek permission at all times. It is stated in the Holy Qur'an: "O you who have believed, let those whom your right hands possess and those who have not reached puberty among you ask permission of you three times: before the prayer of Fajr and when you put aside your garments [for rest] at noon and after the prayer of 'Isha'. [These are] three times of privacy for you. There is no blame upon you nor upon them beyond these [periods], for they circulate among you—some of you upon others. Thus does Allah make clear to you the verses; and Allah is Knowing and Wise. And when the children among you reach puberty, let them ask permission as those before them asked permission. Thus does Allah make clear to you His verses; and Allah is Knowing and Wise." (An-Nur/58-59).

The reason for seeking permission is to cover 'awrahs, and when the reason ceases to exist, the ruling ceases. Currently, with the presence of private bedrooms, the reason for children seeking permission upon entering the house in general has ceased, but the ruling of seeking permission in the bedroom during the three specified times remains valid.

The Prophet (peace be upon him) commanded separating children in their beds when they reach ten years of age. ("Command your children to pray when they are seven years old, and beat them for it when they are ten years old, and separate them in their beds.") Therefore, it is not permissible for a brother and sister to sleep in the same bed after this age. Each must have their own separate bed, even if they are in the same room, while being careful to avoid exposing their 'awrahs.