I won't get into all the evidence that shows the father is obligated to spend on his wife and children since the question asks about the duration.
How long parents should help their children?
Scholars said if son has enough money then the father is not obligated to sponsor him. while they differ on whether the father should sponsor his poor, adult, capable to work son. Majority say he is not obligated to do so.
However, if are not capable of working, then your father must help you.
As for girls who reached puberty, Majority said father should sponsor her until she gets married.
However, that doesn't mean they can't help you. Actually, if they do, they will get double the deeds:
Salman bin 'Amir (May Allah be pleased with him) reported:
The Prophet (ﷺ) said, "When you break fast, you should do it with a date-fruit for there is blessing in it, and if you do not find a date- fruit, break it with water for it is pure." Messenger of Allah (ﷺ) added: "Charity towards a poor person is charity, and towards a relation is both charity and maintaining the ties (of kinship)".
[Tirmidhi] - sunnah.com
My parents say that they have more responsibility toward others (relatives, non-family members etc) and they will not be able to help me any more.
What they are saying is false, although they are not obligated to spend on you. Their first responsibility is to help their parents (your grandparents) then their children (you) then other people in the family. As came in the hadith in Bukhary - sunnah.com:
Abu Hurairah (May Allah be pleased with him) reported:
The Prophet (ﷺ) said, "The upper hand is better than the lower one (i.e., the spending hand is better than the receiving hand); and begin (charity) with those who are under your care;....."
Note: Adoption is not permissible, however, sponsoring a child is allowed. If by adoption you mean the child change his last name to your fathers name and then the child inherits from your father and mother, then this is not allowed.