Good grief. So many people up in arms over a simple venting comment by the OP. 
I'm married now, but was a single parent for years. It is hard work, but so is a relationship where the parenting responsibility is all on one partner. I hardly think the OP was flippantly saying that being a single parent is easy. It sounds like she was merely saying that from a parenting point of view, she's tired of having to be the responsible one all the time, with no help from her partner. Is it REALLY that hard for people to understand where she is coming from? I don't recall having my compassion ripped out when I became a single parent. 
OP, I agree, it can be exhausting. My DH is currently on meds for clinical depression and has a really difficult time dealing with the children right now, especially our DS2 with SNs. He is forgetful and tends to not think things through very well at the moment. This regularly drives me spare as I do feel as if I have no support at all - I have to remember everything, monitor everything, and cannot leave him alone with both children (and often not with either of them, depending on how he is coping any particular day). His counselling is starting soon, thank goodness, or I would probably be sectioned soon. It IS like having another child (or more accurately a stroppy teenager
), and I will admit that aspects of it remind me quite clearly of when I was a single parent, so the comparison is actually IMO fairly accurate.
YANBU to expect your partner to be a responsible adult. If he cannot monitor the children safely while he is on the computer (using this as an example as you did), then I would say you need to discuss the necessity of the computer being off while he is monitoring the children. This is actually a stipulation in our house as well, as DH has difficulty multitasking (yeah, I know, shock that!
) and cannot be online and still keep track of the children safely. So when he is in charge of the children (for example when I'm cleaning in the kitchen or doing the grocery shop or whatever), then he turns off the computer until I'm back. He grumbled initially, but it gets tweaked occasionally depending on which child he is watching and what time of day it is (naptime, etc).