I and my husband are now living separately, and have been for about 4 months now. The reason we argued is because of his wrong doing, and he divorced me in front of both our parents. After a couple of weeks, he ask forgiveness and said that it was not his intention to talaq me. He only said that because he was angry. Now our problem is both our parents already consider the talaq was valid, and they don't even want us to be together again. How do we proceed from here?
if this is your first or second divorce he can get you back as his wife.
if you had 3 "periods" after this divorce you have to get a new marriage contract with your father approval.
if you didn't have 3 "periods" after this divorce he can easily get you back without a new contract or even your permission or presence, he can "preferably" tell two men as his witnesses and that's it. and in this case your parents have no right to forbid it.
about your parents permission, parents are not equal in this matter, only your father has the right to approve marriage, but if he does not approve and you think he has no good justified reason for this you can ignore his permission. being mad of your husband or afraid he would do it again is not a good reason, equality is a good condition to approve marriage and since he approved him before it means equality was not an issue.
so, as i understand, you have contradicting signals: at one hand, you are living separately (i assume you mean also that you are not having sex), in other hand, he said that he did not want to divorce. is saying "i did not want to divorce" counted as taking you back? - i am not sure. maybe he was afraid that 1 divorce can be counted, and for that, wanted to make it not counted at all. but, as he said that after 2 weeks, divorce would not be counted, if he clearly said "i take you back". proof: Just reciting "Talak" thrice commits divorce? . but if you believe that his intention was really taking you back, i think, you can count that as taking you back, and this divorce is not counted at all. but since you are still not living together and seems he had not said clearly that he wants to take you back, it might be some fraud. i see from https://islamqa.info/en/45174 that whether divorce in anger is counted depends on type of anger. i do not know how much he was in anger, but maybe he has approved his intention to divorce by "waiting iddah".
i think whether he was in anger or not is not important, because anger is of short time, and he can take you back when he calms down. and the hadith to which they refer ( i think it is this: https://sunnah.com/urn/1263650 ) to proof anger is important is with word "ighlaq", i have looked that word in lane dictionary and i think that maybe it really even does not mean "anger". and they on islamqa.info also not sure, they say: "The majority of scholars said that ighlaaq means compulsion or anger, i.e., intense anger." , and also translated as "coersion" on sunnah.com, and main meaning of the root on lane dictionary seems to be "to be locked" ( http://www.tyndalearchive.com/tabs/lane/ ).
there is ambiguity in your case, whether you have that divorce counted for aim of counting 3 divorces, after which you cannot reconcile without you marrying other man. i think i can remove/solve that ambiguity, see below.
in any case, you can continue to live together, if this is your 1st divorce, because either he has taken you back before iddah and divorce is not counted, either divorce is counted, but since you are divorcee woman, you can remarry without your parents' consent.
solution of the ambiguity: witnesses are needed to divorce to be counted after iddah is passed, according to quran. so, in your case, you have not formal witnesses, but only your parents. i think, whether you are divorced depends on quality of witnessing of your parents, and whether they can be counted as witnesses - they should be at least 2 just musim men, according to quran. why do they think that your divorce is valid? so, iddah time - 3 periods passed and after then you, together with your husband, said to them, to your father and to his father and others, while they are together, that iddah is passed and your husband has not taken you back in that time? if your fathers cannot witness that, then the divorce is not counted.
in case all was ok with witnesses and they think you are divorced, then: if you say here truth, but lied to your witnesses, i think divorce is counted because they are witnesses; if you said them truth (that he has not taken you back) and you are lying here to us, then of course divorce is counted.
salaams honestly, just make istikhaara about whether or not you should marry him again, and if you get a good sign, then marry him regardless of what anyone thinks, and if not then dont. Just have your heart purely for Allah, and ask for His guidance, as He knows what is best for you in this life and the hereafter I wish everything well between you and your husband, amen