peeing problem!

We have two male neutered cats, I think they are about 10 years old. They have never had any problems using their litter box, even when moving to a new house, taking them to our vacation house, etc. Recently, we installed a cat door to the basement (that's where their litter box is.) My parents tapped it open (they didnt want it to hit them and scare them on the way out) and they also left the door open (as usual) so they could get through that way. They put some food on the other side and shut the door to show them to jump through,but after that they opened it. One cat has no peed 3 times in my parents bedroom (which is across from the basement door.) I am not home (away at school) but they say he doesnt seem to be in pain or anything else unusual. The first two times it was right after they put the door in and the last time was about a week away later (yesterday.) They litter box is clean and nothing else but the door has changed.
My mom also mentioned that there has been a female cat in the yard (they are indoor/outdoor cats) so she was wondering if that could be causing the peeing. They were adopted from a shelter and were both neutered as little kittens before they left!

Forgot to add that after the first two pees the cat vomited up worms. They went directly to the vet and were treated. Not sure if that has anything to do with it .

I just saw the inappropriate peeing thread and will read through that. Sorry I am new here!
Any ideas?

im not sure if worms would make a cat not use his litter box unless he was in any pain. if there was pain then he would have associated the litter box with pain , so that can often lead to them weeing in the wrong places.
it could be a number of things , the change with the door like you said could be one of them , i have just moved 1 of my litter boxs to downstairs with the other ones , but then i had one weeing in the bathroom , since i put it back , no more weeing.
it could be an infection? so a quick vet check might be in order.
or it could be the other cat in the garden that you mentuned. sometimes cats even though neutered will mark there terrertorie, and that dosnt always mean *spraying* it can just sometimes be weeing in places.
so first i would try moving the litter box somewhere other then where it is now or maybe get a second one and put it somewhere different.
i would also take a trip to the vets just in case.
the cat in the garden i dont have any info for that im affraid . maybe buy a feliaway plug in and a spray one?
hope this helps

If there are no health issues the problem is the cat door. If your boys have always had easy access to the boxes, the cat door may have caused some stress and confusion. I even suspect that the door must have accidentally hit them and startle them somehow. You may have to remove the door but if that is not an option, put the box right beside the open cat door. Once they are used to it, close the door but do not move the box. Observe them and see how they react for the next few days. Hopefully they have accepted the changes. Good luck.