Tricky to say - my dad (frail, but no dementia) finds birthdays a bit difficult to remember, but can work it out. My mum otoh, may still know when it is, but if asked can't say it, and couldn't take steps to sort it out.
The memory function of the brain is a somewhat strange thing. I do know that when I was working (Mondays and Thursdays) I instantly knew on waking up what day of the week it was. But ever since I retired a few years ago, that's not a priority, and I often don't know what day it is because I don't need to. (Most shops are open every day, the library is open every day, and so on.)
I think it's often a case of priorities in terms of what people can easily recall through short- and long-term memory. And that does vary throughout the years from the very young to the very old.
But I think CMOT's examples of her parents gives a very good clue.
Memories in the brain are usually stored in the brain in a fragmented way. Some memories aren't easily retrieved, even seemingly important ones, because they don't have a high enough priority - sometimes simply because that person is focussing on other things and is confident that others will remind them.
But sometimes it's because the mechanism for connecting up those fragmented memories isn't working so well.
It could be an issue of cognitive function, it could be a symptom of depression or low energy, it could be an emotional response (your mum is in a nursing home, I think?), it could be an infection as UTIs are notorious for disrupting thinking especially in older people. An assessment by your dad's GP is the starting point to figuring out what's going on.