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. [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.]
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 ones parents.
 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 Kibud Av Vaeim.
 Meiri and Rameh ibid
 Tashbeitz ibid; Peas HaShulchan 2:8 in the opinion of Rashi; Beis Shlomo Y.D. 2:94