Should parents allow their babies to cry themselves to sleep, I would say no, not routinely. It also depends on several factors like: how old the baby is; and why are they crying. There is a big difference between a 21-month-old crying because he doesn't want to go to bed and a one-month-old crying because they have gas pain and can't release it without help. A parent should always check on the child and evaluate the situation each and every time.
Crying is the way babies communicate. They can't say, "Hey ma, my tummy hurts!" or "I'm getting a fever." It is our job to go up and check on them no matter how tired we may be or even if we are frustrated. I frequently checked on my son as an infant and even now as a toddler. Sometimes he is just crabby, other times his teeth hurt and he needs Oragel; other times he needs some Tylenol because he's getting a fever. Other times yet, he's wet or pooped his diaper- would you want to sleep in that? I wouldn't - don't blame him a bit! Kids cry for all sorts of reasons and to stubbornly ignore their cries is wrong. You could call it discipline, but at certain ages discipline isn't valid.
There are lots of things you can do if your baby is crying. Try burping them; Check their diaper. Maybe even feed them a bit - especially if they're on breastmilk. It could be they are either too warm or cold. Go down the list. Sometimes a simple check and fix is enough to put them back to sleep. There are times when a child will cry because of a temper tantrum, but that's usally more toward the toddler years. Young infants are usually very honest about their needs.
Child development studies have shown that babies of parents who respond to their cries feel safer and more secure and are more well-adjusted than ones whose parents routinely ignore them. Who wants their kid growing up feeling unloved or neglected? Soon your baby will be able to remember things. Do you want him to remember when he had a tummy-ache and you didn't help?
Even with the professional data aside, as a parent it is your job and your duty to check on the child. You never know if they've got their leg stuck in the crib or if they're suddenly spiked a fever. To leave a child crying and not try to help them or understand what is the matter, is wrong and bad parenting, especially if you are dealing with a young infant who is almost 100% helpless. I'm not saying you need to coddle your child all of the time, but the obvious message in this article is: go, check and evaluate the situation when your baby cries. They normally have a good reason.