The Shulhan Aruch, in the Yore De’a section (240:7), codifies the Torah obligation to stand in the presence of one’s father or mother. The source for this Halacha is a passage in Masechet Kiddushin (31b), where we read that Rav Yosef would say upon hearing his mother’s footsteps, "I shall rise for the Shechina [Divine Presence] which is now coming!"
The Rambam (Rabbi Moshe Maimonides, Spain-Egypt, 1135-1204), in codifying this Halacha (Hilchot Mamrim 6:3; listen to audio recording for precise citation), writes that one must stand in the presence of one’s parent "as he stands in the presence of his Rabbi." The later scholars interpret the Rambam’s comment to mean that this obligation is akin to the obligation to rise before one’s "Rab Mubhak" – his primary Torah mentor. Halacha requires standing in the presence of one’s "Rab Mubhak" so long as he can be seen; even if the Rabbi is not situated near the student, the student must rise if the Rabbi is within visible distance. Accordingly, in the presence of a parent, too, one must stand even if the parent is at a distance. So long as the child can see his father or mother, he must stand as an expression of honor.
This obligation applies regardless of the parent’s age, and regardless of the parent’s level of Torah knowledge and piety.
Rabbi Moshe Feinstein (Russia-New York, 1895-1986), in one of his responsa, bemoans the fact that so many people neglect this Torah obligation. Some suggested that nowadays it can be assumed that parents forego on this expression of honor, and do not demand that their children stand in their presence. Since a child is not required to stand if the parents waived this obligation, nowadays people need not stand, as they can assume that their parents forego on this requirement. However, children who wish to rely on this leniency should receive explicit, verbal consent from their parents not to stand in their presence, so that it is clear that they forego on this display of respect which is due to them.
Summary: There is a Torah obligation to stand in the presence of one’s father or mother, so long as the parent is within visible distance. Some authorities maintain that nowadays it can be assumed that parents waive this privilege; however, it is proper for one to receive their parents’ explicit, verbal consent if he wishes to rely on this leniency.