Traveling from Eretz Yisrael for the sake of one’s parents:
Although in general it is forbidden for a resident of Eretz Yisrael to leave Eretz Yisrael, it is permitted for one to leave Eretz Yisrael for the sake of visiting his parents or caring for his parents. [This applies even to a Kohen. It goes without saying that if one’s parents are traveling from Eretz Yisrael for a certain medical or financial need, then it is permitted for one to travel from Eretz Yisrael together with one’s parents in order to escort them and assist them in their journey. However, one may not leave Israel for the sake of meeting the body of a parent which is on its way for burial in Israel.]
Moving to the Diaspora for the sake of one’s parents: Some Poskim rule that it is only permitted for one to temporarily leave Eretz Yisrael for the sake of visiting and caring for his parents, however, he may not do so for permanent basis. Other Poskim, however, rule that it is permitted to move to the diaspora even permanently for the sake of caring for one’s parents.
 Rameh and Meiri and Hamakneh on Kiddushin 31b; Tashbeitz 3:288; Maharsha Chidushei Agados Kiddushin 31b; Minchas Chinuch Mitzvah 212; Pesakim Teshuvos 240:46; See Kiddushin 31b “Rav Assi had an elderly mother and would do whatever she asked. If she asked for jewelry, he would give her jewelry. When she asked for a husband, he replied that he would search for one on her behalf. When however, she asked for a husband like him, he realized that she lost her mind, and he therefore left her for Israel. When he heard that she was coming to see him in Israel he approached Rebbe Yochanon and asked him if he may leave Israel to the Diaspora? He replied that it is forbidden to leave Israel. So, he asked him if it is permitted to leave Israel in order to greet his mother? He replied that he does not know. After some time, Rav Assi asked Rebbe Yochanon a second time the same question, to which he replied that since he sees that he really wants to leave Israel, may God return him to Babylon in peace. Rav Assi approached Rebbe Elazar asked him if Rebbe Yochanon was angry with him and therefore answered him in this manner. Rebbe Elazar replied that since Rebbe Yochanon blessed him, he was therefore serious in his answer not angry. In the end, his mother passed away after he left Israel to greet her, and Rav Assi exclaimed that if he would have known he would have not left Israel.”; Encyclopedia Talmudit ibid p. 439 footnotes 494
 See Michaber O.C. 511:6; Y.D. 372:1; Rambam Melachim 5:9; Kesubos 11a; Avoda Zara 13a; Kiddushin 31b; M”B 511:14; Birkeiy Yosef 568; Peri Haretz E.H. 3:11; Shaareiy Teshuvah 568:20; Sdei Chemed Eretz Yisrael 37; Nitei Gavriel 90:4; Likkutei Sichos 15:202 footnote 15; 18:399; 25:151; Shulchan Menachem 7:147-148; Sichos Kodesh 5716 34, 2nd day of Chag Shavuos p. 276; Sichas night of Gimel Tamuz p. 290; Igros Kodesh 18:404; Gilyon Kefar Chabad 93 p. 12; 86
 The reason: The reason behind all this is because it is permitted for one to leave Eretz Yisrael for the sake of fulfilling a mitzvah, and by visiting one’s parents and certainly by caring for them in the diaspora, one fulfills the Mitzvah of Kibbud Av Vaeim.
 Poskim in Encyclopedia Talmudit ibid footnote 906
 See Noda Beyehuda Tinyana E.H. 45; Encyclopedia Talmudit Erech Kibud Av Vaeim Vol. 26 footnote 1089-1095
 Meiri and Rameh ibid
 Tashbeitz ibid; Peas Hashulchan 2:8 in the opinion of Rashi; Beis Shlomo Y.D. 2:94