That MIGHT hold for people who are sensitive for others, i.e. those with lots of empathy, but I believe that the OP meant people who would generally be considered as "butthurt", or too concerned about themselves.
And I agree that such people are plague, although they are not necessarily raised as insensitive, I think they're often just used to abusing their parents' love and then they are hurt when they find out others won't tolerate their sh** as much as parents do, or won't talk to them as nicely.

That's all people though, not just young people.

What parents should be doing is exposing their children to as many different kinds of people as possible when they're young so that they learn tact which imo is the most important thing anyone can learn. Being able to deal with people, and knowing what to say and who you can say certain things to and when is the most important thing, and will ultimately lead to success in whatever career path they choose.