Praise be to Allah.
Firstly:
If a person does not fast because of an excuse such as travel or sickness from which he hopes to recover, then he has to make it up. If he dies without making it up, even though he was able to do so, then it remains an obligation that must be fulfilled, and it is mustahabb for his next-of-kin to fast on his behalf, because of the hadeeth of ‘Aa’ishah (may Allah be pleased with her), according to which the Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever dies owing any (obligatory) fasts, his next-of-kin may make them up on his behalf.”
Narrated by al-Bukhaari, (1952), Muslim (1147).
But if he dies before he is able to make them up, such as one whose illness lasts until he dies, then he does not have to do anything and his next-of-kin do not have to do anything on his behalf.
If a person does not fast out of carelessness and heedlessness, and has no excuse, he does not have to make them up and they are not valid if he does, because the time for them has ended.
This has been discussed previously in the answer to question no. 50067 and 81030
Based on that:
What appears to be the case with your father, as he was keen to pray and do good, is that he would not have refrained from fasting without an excuse, so it seems that he refrained from fasting because of the excuse of travel, and it is not known whether he made up those days whilst travelling or in the winter, for example -- and your mother does not know either -- or whether he did not make them up. Was it possible for him to make them up when he had a break from work and stayed home, or was he always travelling because of the nature of his work, so that he was not able to make up what he had missed until he died?
In the face of these possibilities, we say: if you cannot find out the facts of the matter, and you fast on his behalf as much as you are able to, this is a good and righteous deed for which you will be rewarded in sha Allah. But it is not obligatory and you do not have to find out the exact number of years he did not fast; rather you should act upon what is most likely to be the case and decide on the number of years that he did not fast, then fast on his behalf as much as you can. This comes under the heading of ihsaan (kind deeds), but it should not distract you from other deeds that are more important and more beneficial.
It is permissible for all the heirs to join in this making up of fasts. Whatever it is too hard for them to fast, they may feed poor persons instead, one poor person for each day.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: It is mustahabb for his next-of-kin to make it up if he did not do it. We say: for each day, feed one poor person.
And he said: If we assume that a man has fifteen sons, and each of them wants to fast two days to make up for thirty days, that is acceptable. If there were thirty heirs and they all fasted one day each, that would be acceptable, because they fasted thirty days between them. It makes no difference whether they all fast on the same day or one of them fasts one day, then the next fasts the next day, and so on until they complete thirty days.
End quote from ash-Sharh al-Mumti‘, 6/450-452
Secondly:
With regard to the fasts your mother did not observe from after she reached puberty and until her marriage, this matter is subject to further discussion, as follows:
1-Whatever she did not do out of heedlessness and carelessness, with no excuse, she does not have to make up, as stated above.
2-Whatever she did not do because of an excuse such as menses, travel or sickness, she has to make up. She should try hard to work out the number of days or what she thinks is most likely.
And Allah knows best.
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