I wouldn't describe it as a "lie" so much as challenging the socially prescribed link between sex and gender. Trans people know they're biologically male, and they know you probably know it. But they're challenging the social expectations that get associated with that.

The same analogy can again be drawn with homosexuality. 100 years ago, many people believed that it was a biological fact that men preferred women and vice versa. To hear a man proclaiming that he prefers men, or a women proclaiming she prefers women, sounded to them like a lie. But as society progresses, social norms tend to relax. In fact there are people to this day who cannot quite accept the idea that homosexuals might genuinely exist. To such people, seeing a gay couple kissing feels like being expected to ignore a biological fact.

The crux of the problem is that the parent believes that putting a trans "woman" on a product that is intended for people who never asked for it is supposed to spread tolerance and acceptance.

The parent's view is that it is the fault of the consumer that voted against such imagery because they do not identify with its messaging because they were "nazis"

Yet seems oblivious to the fact that if it was done on to their favorite product, they would react in the exact similar manner. For instance, trying to stop the sales of a game which depicts women in a voluptuous manner, purposefully making them more "butch" because it offends a small group of people.

This is the exact issue that we recognize as wokeness that someone on the far left is incapable of tolerating. So the issue is always one side wants to force tolerance by forceful means, the other just wants to be left alone and leave it to individual tolerance.

The collective brute force method has completely backfired yet this still does not register in the mind of those that still think indoctrinating, forcing ideology on to people is going to lead to more tolerance.