If you want to be proactive, and it sounds like your parents may be newbie dog owners... here are some tips, for 'in case they show up with dogs' this year.. and for next year/subsequent family events.
Emergency situation - dogs have to visit (you never know, better to be prepared).
Cat into an upstairs room with litterbox, food, etc.
Dogs in kitchen (closest access to outdoors usually).
This should in most houses give you TWO doors between dogs and cat.
Feliway plug in/collar/spray for cat room.
Dap plug in/collars/spray for dogs in kitchen.
If there are small children, then SERIOUSLY consider fitting some sort of temporary child proof lock to kitchen door so children CANNOT get in with dogs unsupervised or let dogs out (Because frankly, if they can, they will).
IF your parents have to bring the dogs insist that the dogs are crate trained (shouldn't take more than two weeks for them to train them sufficiently to use the crates in someone elses home though this WON'T mean they can be crated AND left unsupervised, thats likely to take a month or so as it involves gradually building up time) or trained to settle on a mat and not move until told to (I fancy crate training is the better option!).
I don't think you should have to have your parents dogs in your home if you do not want them but if you can tell your parents 'unless you crate train and supervise them whilst they are here then they cannot come' you sound much more reasonable - and if they DO crate train them then they won't be a bother - and you still have the cat as a good reason for dogs not to visit unless a dire emergency ('we forgot to book a kennel space/dog sitter is not a dire emergency!)..
Otherwise its likely that once you no longer have that cat, they will assume the dogs can come as it was the cat that was the reason!
-Find dog friendly b&b's local to you - check that they will allow dogs to be left there unsupervised (many won't. Some will, if dogs can be crated, some offer a dog sitting service).
-Find dog kennels local to you - check how far in advance they recommend you book for. Suggest to your parents they visit these places next time they are with you for suitability (NEVER leave a dog in kennels that you've never visited!)
- Recommend they crate train their dogs and other useful skills (toilet on cue, settle on a mat and leave it are three easy to teach and highly useful skills that make a dog far less irritating a visitor!)
These are just suggestions though to pre-empt stuff in future, in the end it is your home and your choice who or what enters it.
I don't have children in my home due to my dogs and thats not a rule I'll ever contemplate breaking, no matter how much it may offend or how awkward it makes social or family events.