Can one parent have a disease trait, and the child inherit the full disease?

What disease. Lets say it is sickle cell trait. You must have 2 defective gene copies ie an abnormal one from both parents to have the full blown disease. Both parents must be carriers and it depends on what disease you are referring to.
Not without help. One parent with a "trait" or asymptomatic condition cannot give the full condition to an offspring. For the child to have a full disease, the other parent must either be a carrier or have the disease and pass that half of the gene to the kid. Parents who are minimally symptomatic with a dominant disease can have a kid with more symptoms, but this is never called a trait.