Praise be to Allah.
If a person is not sure of the months, that does not mean that the fast of Ramadan is waived in his case; rather he must try hard and do his best to find out what month it is.
It says in al-Mawsoo‘ah al-Fiqhiyyah (10/192):
If a person is in prison or he is in some remote place far away from the cities, or he is in hostile territory where he cannot find out about the months from other people, and he is not sure about when Ramadan is, then in this case the fuqaha’ are agreed that he must try hard and do his best to figure out when Ramadan is, because it is possible to fulfil his obligation by working it out and doing his best, therefore he is obliged to do that, as in the case of facing towards the qiblah. End quote.
If he tries hard to work out the correct time for fasting, then his worship will be valid and will be acceptable, because Allah, may He be exalted, says (interpretation of the meaning):
“On no soul does Allah place a burden greater than it can bear”
“Allah puts no burden on any person beyond what He has given him”
Please see also the answer to question no. 81421
So what your father should have done was try hard to work out when the month was and to fast according to what he thought was the case. If he could have asked someone, then he should have done so.
When he found out that Ramadan had begun, or had ended, then he should have fasted, either by doing it if he found out at the time, or making it up if the month had already ended.
As for staying for two years without fasting and giving the excuse that he did not know whether the month had begun, that is not permissible.
Your father must fast for two months in place of the two Ramadans when he did not fast; he must also repent, ask for forgiveness, and do a lot of supererogatory (naafil) righteous deeds, especially fasting.
In fact the majority of scholars are of the view that he must, in addition to making up the days when he did not fast, feed one poor person for each day that he did not fast.
Shaykh Ibn Jibreen (may Allah have mercy on him) was asked:
If someone delayed making up missed Ramadan fasts until the next Ramadan, what must he do?
If that was for an excuse, such as if he was sick in bed for eleven months and was not able to fast during this period, then all he has to do is make up the missed fasts. But if it was negligence and heedlessness on his part, and he was able to do it, then in addition to making up the fasts he must also feed one poor person for each day, as expiation for carelessness.
End quote from Fataawa as-Siyaam
Please see also the answer to question no. 26865
If a person is unable to make up the fasts because of sickness or old age, then what he must do, in addition to repenting, is feed one poor person for each day. The majority of scholars say that he must also feed another poor person for each day as expiation for the delay in making up the missed fasts.
Jalaal ad-Deen al-Mahalli (may Allah have mercy on him) said in his commentary on Minhaaj at-Taalibeen (2/88):
The more correct view is that if he delays making up the fasts even though he is able to do so, then he dies, then two mudds must be taken from his estate for each day, one for having missed the fast and one for the delay.
The second view is that one mudd is sufficient, which is for having missed the fast, but the mudd for the delay is waived. End quote.
Therefore if he is able to feed two poor persons for each day, that is more prudent and will ensure that he has discharged his obligation, otherwise he may feed one poor person for each day, and he does not have to do anything besides that.
And Allah knows best.