I was brought up in a midwest family where "a lot" meant a piece of ground, and not "many". Different generation grammar usage?...No doubt, the parents thought they were "protecting" the language (or perhaps the children). There is, however, absolutely no grounds for this except the false idea that words have only a single sense. Clearly they don't. The Longman Dictionary of Contemporary English gives 31 senses of work and that's just the verb. Lot similarly, if less spectacularly, has a number of common senses; I wonder what the father would have called a parcel of articles offered as one item in an auction? Can, too, has multiple uses, including ability, requests, possibility, and permission.
We also learned that "Can I go to the store?" brought the answer "Yes, you can, ...but you may not."
Merriam Webster's has this to say:
"Can and may are most frequently interchangeable in senses denoting possibility; because the possibility of one's doing something may depend on another's acquiescence, they have also become interchangeable in the sense denoting permission. The use of can to ask or grant permission has been common since the 19th century and is well established, although some commentators feel may is more appropriate in formal contexts. May is relatively rare in negative constructions (mayn't is not common); cannot and can't are usual in such contexts."So, is it a generational thing? Only in the sense that younger people rarely think to inflict their linguistic preferences on others. But if the question is whether this has changed, indeed, it has, just not within the lifetime of the father. Lot in the sense of a large number goes back to at least 1812 which is not all that long after it was first attested in the meaning that the father prefers.