I don't think it's so much that they think that LGBT means pedo. It's a worry that the reactionary protection of all things LGBTQ will provide cover. Many of the slogans and explanations provided for pride could also apply to pedophilia. I.e. "love is love" or "the attraction is natural". Of course there is a big difference: adult consent.
This is why there is a lot of concern when minors are dragged into LGBTQ issues. They aren't old enough to understand or especially to consent to being exposed to such issues. What it is okay to expose them to is hotly debated. And the reactionary protection of all things perceived LGBTQ can provide cover for a lot of questionable activities. For instance, underage drag shows at adult clubs. If a cis-gendered minor performed at an adult only club, it would definitely be denounced as a sexualizing of the child. But because the performance involves cross dressing it's now a healthy expression of their non-conforming sexuality? No.
Any form of sexuality still need's to be expressed and discussed in an age appropriate manner. Too often that is not what is happening. Unfortunately, that also isn't what's being addressed by either side. The LGBTQ side is so eager for acceptance they want almost everything related to be considered acceptable, while on the flip side, some parents are so concerned with what might smuggle itself through under the guise of LGBT acceptance that they want everything blocked.
Essentially It's why I would never take a child to a gay pride parade. It's not that I have a problem with gay couples, appropriate pda (holding hands and non make out kissing) or rainbows, it's the bondage gear and d**dos on display. And those aren't even gay, straight people are into that too, but it doesn't belong on public display.