A true parent never gives up. But you have to clarify what that means. It does not mean that a parent should enable a child to continue to harm himself or others. It means that a parent will always do whatever it takes to help his child receive the help he needs to live a safe, healthy and happy existence as much as possible. When a child, even an adult child, obviously struggles with psychological challenges, he could never just have him locked up and throw away the key. A true parent would be sure to provide the necessary medical, psychological, social and spiritual support. Some people have difficult states of mind that make it tough to relate to them, but there are pockets of ability within that should be recognized, salvaged...valued.
I'm not saying that a parent should take on the full-time care-taking of such a child but perhaps quality residential treatment including therapy and medication would make it easier to maintain loving family relationships.
There is so much misunderstanding about people with mental disorders. Those around them need to separate the person from the affliction as much as possible. Love the person, treat the affliction, safeguard yourself from harm, connect in safe ways.
Parents often feel like failures if their child grows to face a hard life. The only failure is a failure to understand that seriously anti-social or self-defeating behaviors stem from brain problems. It's not a personal rejection or judgment of one's parenting skills or worth as a human being. Everyone needs love. A true parent's love is unconditional even if it means it can't be understood or returned by his child.