Marriage vows have different wordings but explicitly said in some and worded differently in others, is the phrase '^forsaking all others^ and 'man and wife become 1'.
You may not have gone through a marriage ceremony but the length of your partnership and the existence of children means, de facto, you are married.
The marriage service was always very clear that when you married the link between parent and child is severed and that your main responsibilities are to each other, and your parents detailed responsibility and control of your life ends.
The problem is that your parents are not fulfilling their part of the bargain. They are still treating you as if you were their child in whose life they can continue to interfere on a daily basis. Any parent who says as your mother did that she did not want the father's name on the birth cirtificate and the children to bear his name is so way way way over any line, that you should have called her out then and there, not forgiven her.
As for your father asking now 12? 15? years into your partnership 'Did you think he (the grandfather) would let anyone take his grandchildren' leaves me speechless. What on earth is he talking about? Does he see your partner's family trying to run off with them, does he think your partner is about to leave you? Children have 2 biological parents, doesyour father think he has some right over the children that your partner does not? Your father has absolutely no right to make any suggestion such as he made. He is as far over any acceptable boundary as his wife.
There is no question of being estranged from your parents or your parents not seeing the grandchildren or of you not loving your parents. It is the simple fact that when you and your partner formed your partnership, from which you have children, your parents should have taken a step back and accepted that you were now living a life where your first responsibility was to your partner and children and that in their dealings with you they had to deal de facto with the whole household, not you as an independent person on your own.
What you have got to do is make sure your parents understand where the boundaries are in their relationship with your partner and children. make clear that you and your family are as valid a family unit as any couple who have married. That they have no right to question any decisions you and your partner make about your family and they are not to say anything to you that explicitly or implicitly criticises your partner, casts doubts on his commitment to you and his family, or suggests that he or any member of his family wishes to do anything to undermine your family unit.
You can best choose how to do this, face to face with your partner or in a joint letter. Assure them of your love and talk about how much your children love their grandparents and enjoy being with them and then make absolutely clear where the boundaries are in your family's relationship with them and make it clear you will not accept any unwarranted interference in your joint lives or any suggestions that your partner and his family are untrustworthy.
If they do offend then both of you together should confront them with their accusations and ask them to justify them.