Basically, both of my parents are blood type A (both are confirmed and it's also certain both of them are my biological parents). I recently found out my blood type is AB. How is this possible? I tried reading about different mutations in blood type, but I couldn't find anything that was similar to my case.
Blood type A has a variety of subtypes, which have different features in terms of their antigens. This leads to people with different A subtypes having somewhat different kinds of antibodies in their blood (for a technical article, see here). I don't know the population prevalences well, but I read somewhere (can't find it now) that as many as 20% of blood type A people actually have one of the subtypes. And these are themselves going through their own mutations. It is possible that you have an A subtype of some kind.
Testing blood types usually consists of testing which antigens and antibodies are present in your blood (see e.g. here). So if you have some kind of A subtype that didn't fit well into the blood test's expectations, you might have shown up as AB. I'm not a hematologist or physician, so I'm not sure how plausible that is.
But if you are really interested, you might consider getting a DNA test that will identify which type/sub-type you have, as this will add a different source of information on top of the standard blood test. In the jargon, the standard blood test is for "phenotype" (what are your blood cells actually doing), whereas DNA tests are for "genotype" (what is your DNA sequence). Often genotype and phenotype have less to do with each other than you might expect, or they don't fit in all that well with the cut-and-dried tests. In particular, a lot of medically underserved populations have variation that isn't really accounted for by standard medical typology.
Hope that helps.