I'm so sorry you have to give up your boys! But it is truly wonderful of you to care for your father.
Obviously I don't know anything about your father's illness, but are you sure you have to give up the cats? Is the reason for giving up the cats an allergy that exacerbates an existing breathing problem? I won't assume that you have already explored allergy problem alternatives. I don't know if your father is willing or in a position to take allergy medication. But just in case you haven't thought of this (though you probably have), there are alternatives if this is the problem. I am VERY
allergic to cats. Yet I live in a small home with five of them. But I'm willing to take Zyrtec-D twice a day, Advair twice a day, carry an albuterol inhaler, and keep steroid cream around for when I get hives from them licking me. This is extreme - but, like I said, I'm very allergic to cats. Most people with allergies do just fine with either Zyrtec or Zyrtec-D. If you haven't explored these as an option, please look into it.
If you have no choice but to rehome your kitties (God it must be heartbreaking! I'm so sorry you have to go through this), please take Katie's advice. If there is a problem (or perceived problem) with the kitties in their new home, new owners are FAR more likely to relinquish to a shelter (or just dump somewhere) cats (or dogs) acquired for no cost. They have to make no commitment to acquire them, so it's that much easier to relinquish them.
I also highly urge you to use an adoption agreement. Please use the link in my signature line - it'll take you to one you can customize. This was designed for rescue rehoming situations, where kittens are placed before they've been spayed/neutered. With your kids being a year old, I'm assuming they're already neutered. If not - first of all shame on you, but second of all, please take this opportunity to require that they be neutered, and include that in the adoption agreement. Anyone seriously interested in caring for the boys you love will be happy and willing to sign that they agree to your requirements for a new home/family.
Finally, is it possible to keep the boys confined in a separate room, bedroom or basement while you find them a new home? That would "buy you some time," so to speak, and not leave you with no choice but to give them up to the Humane Society when your father arrives. Of course they wouldn't be happy, having had free roam of the home before, but if you take the time to create a quality space for them, it would gain you time you need and it would hopefully mean you can screen for a wonderful new family for them. Unfortunately, rehoming cats usually takes several weeks, not several days.
(Unless you have friends who will take them).
Please also consider printing out both parts of this article about bringing new cats home and providing it to your boys' adopter(s). http://www.thecatsite.com/cat_care/care.php?a=newcat
Many people expect to bring cats home and immediately have them be a loving part of the household - when, in fact, the cats are terrified. A little education can go a long way in making everyone happy in the end.
If you are forced to have them out of the house in two days time, please search for no-kill shelters as opposed to the humane society. Here's a link where you can search:http://www.saveourstrays.com/no-kill.htm
Also please just look in your yellow pages and call the shelters near you - some of them may be no kill and not listed in the searchable database. Also try calling your vet - or all local area vets. Those that get a lot of traffic often have people looking to adopt animals. Our vet adopts out three to 15 animals a week. They also know if anyone recently had an animal that passed away and wants or would be willing to take cats in need.