Not sure what size parts are "small" to you, but if they are being cleaned in the house I would presume pretty small and not terribly messy. As of late, I have been tinkering with watches, and as you might imagine, those parts are small. I wont go into the whole "proper watch parts washer" thing, but here is what I use for cleaning small, but not incredibly tiny parts.
The parts are cleaned in very small tubs (think small like margarine, cream cheese, etc), generally with toothbrushes, soap and water, etc.
That tub is contained in a larger tub/vessel, just in case parts jump, or float on the water.
The above 2 are contained in a clean oil drain pan.
The drain pan goes in the sink and used for overflow. The two smaller tubs are clear, so I can see parts.
All this precaution was the result of a recent watch cleaning episode where when I got to re-assembling the band, I was missing the 2 end links that go against the watch face from the band. I looked all over my workbench and didnt find them. I then thought to check the sink, and THANKFULLY they were in the sink strainer, where they would have for sure gone down the drain.
Even then, I didnt know if I had lost them from the trip from the workbench to the kitchen sink.
Hope that helps.