A. Zuhayli in al-Fiqh al-Islami wa adillatuh does not mention staying in your parents' house, nor does he mention staying in your town of birth, as criteria that cancel one's traveller-status in any school.
The Four Schools' agreed-upon criterion that establish such status is, generally speaking: travel a one-way distance of 55 miles [89 km] or more. And Ibn Qudama has argued that even less than that may constitute travel, on the strength of Anas's narration that the Prophet would shorten his prayer if he went out for a distance of 5 or -- one of the narrators does not remember -- 10 miles [8-17 km]. It is narrated by Muslim, Ahmad, and Abu Dawud. Ibn Qudama said (al-Mughni 2:257) that this constitutes enough proof unless there is a consensus to the contrary. Shafi`is fix it at 50 miles [81 km].
There are other conditions as well, but they are not agreed upon accross the school. However, in no school does staying in one's parents' house or one's town of birth cancel this status once met.
Among what cancels the traveller-status is the intention to stay at the place of arrival for a period of fifteen days or more [Hanafi school] or four days or more [Malikis and Shafi`is], or more than four days [Hanbalis], while Malikis and Hanbalis also stipulate "or more than twenty rak`at of prayer in total." (Zuhayli 2:326-327). One does not count the day of arrival nor the day of departure.
Someone said: "The shaf'i opinion is that you are not a traveller once you enter your town of BIRTH, not necessarily your parents' house."
But the words "town of birth" in the sentence above are a mistranslation of the Arabic "watan" -- place of residence -- where the Shafi`is say: travel ends when he reaches his watan. Lisan al-`arab defines watan as: al-manzilu tuqimu bihi, "the place where you reside." So the sense is not that the trip ends when one reaches one's place of birth (as he may have emigrated from it and so he may be a traveller when visiting it again), but only when one reaches one's hometown where he currently resides. That is what you will find in the Shafi`i manual Reliance of the Traveller (p. 192 f15.7) and that is what Zuhayli has also (2:330) for the Shafi`i stipulation, respectively:
wa yantahi bi wusulihi ila watanihi (Reliance)
"It ends when he reaches his hometown."
wa qala al-shafi`iyya... yantahi al-safaru bi wusulihi sur watanihi aw `umranihi (Zuhayli)
The Shafi`is said: travel terminates upon his reaching the precinct-wall of his hometown or its constructions.
Zuhayli also mentions (2:319) the hadith from `A'isha whereby: "I went on `Umra with the Prophet during Ramadan, he broke his fast whereas I fasted, and he shortened his prayer whereas I did not shorten it." Daraqutni narrated it and he said its chain is fair (hasan). Shawkani cites it in Nayl al-awtar (3:202). This hadith shows that the Prophet considered himself a traveller in `Umra although Mecca is his town of birth. (`A'isha's action shows that it is not required to shorten prayer while travelling, as per the view of the Three, and the Hanafis consider it wajib.)
Note: The precise word for "place of birth" in Arabic is mawlid, which also applies to "time of birth" or "birthday." wallahu a`lam.
A dear brother who follows the Hanafi school contradicted the statement that passing through one's birth-place automatically cancels traveller-status. These are his exact words:
i would like to make a point about your answer on traveller's salat. there are mainly 2 types of watan in fiqh: watan asli, watan ikama. the former is where you consider yourself from. for instance my watan asli is istanbul. the second is where you live more than 15 days. for instance my watan iqama is NY because i study here. my parents also live in istanbul. but the fact that your parents live somewhere has no importance with respect to watan. where do you consider yourself from? this determines your watan asli. where do you presently reside? this determines your watan iqama. there can be more than 1 watan iqama. regardless to the amount of time of stay, you're always considered a muqim when you're in your watan asli or iqama. so if i go istanbul even for an hour i'm muqim. likewise new york. if i stop by in new york for an hour on my way to some other city i become immediately muqim.
and the third type of watan is watan safar, you might call. it's the place where you stay less than 15 days or you stay there as a soldier or as a slave or without intending to stay more than 15 days.