Switched insurance, getting monthy bill early from the new insurance, and still getting monthly bills by old insurance…

So I’m going to squish 2 things into this post but both are a similar question.

First of all I’m from a state that uses it’s own exchange. I’m not too worried about my health insurance stuff, just private insurance no financial aid. I just got a monthly bill for my new insurance, but I don’t start the new coverage until January. It seems early to be getting it, usually when I get billed it’s a couple weeks prior to the start, but it’s over a month away, is it normal to get billed early?

This question is for my parents, they’re both on medicare now. One is just starting and the other has been on it for a while. The one just starting, had signed up for a medicare health insurance plan but recently had it changed again before starting it cause they found a better one and my other parent decided to switch too. That last part happened about a month ago, but they’re still receiving monthly bills from their old insurances. They talked to our insurance agent several times about it and supposedly took care of it (more then once) but the bills are still coming through. I assume that for some reason the changes hadn’t yet gone through their system (on the insurance companies’s’ end)? You’d think it would’ve by now though, it’s been at least a month since the most recent changes.