It's the tray in your laundry room where the washing machine empties, where you wash your hands or brush your teeth if you have no bathroom basin, bathe the dog, place your draining plants after you water them, wash your hair if it reaches below your hips...stuff like that.
You can bathe the dog there? wash your hair? I've never, ever seen anything like that in any of the U.S. home laundry setups. Usually the washing machine is directly hooked to a drain pipe of some sort.
My apartment building used to have the washers empty into the slop sink, but when we had things tweaked, our plumber insisted we eliminate that and hook up directly to the drain pipe.
The only "tray" I've ever seen in a laundry room is the overflow tray on the floor, which is about the size of the washer.
Are you in the U.S.?
And sorry, I'm not walking all the way to the washing machine (basement, maybe?) to spit, should I find myself needing to.
Alias, I think your parents' reaction was one of those "this grosses me out" reactions, not some universal rule of Etiquette. (except that I'd say Etiquette would tell you not to spit in someone ELSE's kitchen sink, and in fact to not spit in front of other people in general)
There isn't any logical reason not to do it in your own kitchen. But if it squicks someone out, that's OK too--but they shouldn't act as though their own personal reaction indicates anything about the world in general.
I'm in the US and have never lived in a house without an laundry tray, which is in the laundry room (or wash room, as we usually call it). Our machine drains into the tray with a large hard rubber drain hose that has a big bend it (the bend sits on the edge of the tray). I thought everyone had these.
I had one that appeared to be out of cement once, but the one I have now is fiberglass, on a wooden stand. I love it. Someone said they wish they had one. I am so glad that I do. I use it for so much stuff.
I've only lived in an apartment once and had no laundry facilities at all. Otherwise I've lived in houses with the wash room off of the kitchen. Just goes to show the assumptions we make about how other people live. I don't think it's a rural west coast thing, or is it? I'm feeling kind of quaint, right now.