Are you hurting a cow by milking her?
The short answer is no.
The long answer is still no. But with an explanation as to why.
A cow's udder is a sensitive organ that is designed so that her calf can latch on with his mouth and get that highly nutritious opaque, white body fluid we call "milk" from her. It is quite like a woman's breasts, which also fill up with milk soon after birthing.
So just an aside: Cows, and all female mammals, produce milk soon after birth. So in order to produce milk, they must give birth, not be pregnant.
Now, just imagine that sensitive organ filling up with milk to the point where it literally looks swollen. Again, this is a sensitive organ we're talking about here. When a sensitive organ is swollen for whatever reason, it gets painful.
Think of a more disgusting situation: You have an abscess or boil that is pus-filled on the inside of your leg. Is that going to be a painful boil? I sure hope you said yes because if you said no I'd be worried!
So when that cow's udder keeps producing milk and there's no calf or human hands or milk machine to relieve that painful pressure that keeps building and building, it is indeed painful and quite uncomfortable for that cow. Just like that boil between your legs: It gets so full of pus that it's painful, and the only way to relieve it is to release that pus inside.
Now, let me get this straight: Milking out a cow is FAR less disgusting than lancing a boil or abscess. And the resulting fluid is quite a bit more nourishing!!
Another aside: Milk doesn't have pus in it. If it did, no calf would be able to subsist well on it.
When a cow is milked out, that is a relief for her. It makes the pain go away, and she feels much better. Cows always like it when things happen to them that make them feel safe and happy, and pain-free. Regular milking is associated with this particular feeling.
Cattle are always going to go toward a source that makes them feel good. This is why dairy cows don't need to be made to go to milking. Rather, they are waiting at the gate and eager to get into the milking parlour, before the people who are going to milk them out are even ready to do so.
Let me put it again this way: When something positive and rewarding that gives or means comfort and not painful, animals seek out that source or point in time again and again.
Cows get accustomed to regular milking because a) they’re creatures of habit and like it when things are kept on a timely schedule, no matter if it’s feeding, milking, whatever; and b) milking, as a pain and stress-reliever, becomes something cows quickly learn as a positive reward; all animals, including us humans (normally), don’t like things that cause pain and discomfort, and once something painful and discomforting has gone, we–or they–do our/their best to avoid it happening again.
So since milking is in fact associated with a good-feeling thing for the cows, they are eagerly (and albeit patiently) awaiting the workers or farmer at the gate to be let in for milking time.
But overall, milking is not painful for cows. It would only appear so because, for young cows, having a milk machine attached to their teats is foreign and something they need to get used to.