These two concepts come into conflict because the idea of due process is based on the idea that individual rights must be upheld while the concept of parens patriae is a much more patriarchal vision that does not pay attention to rights.
Under the concept of parens patriae, the state takes over the role of the parent. It does what it thinks is best for the juvenile offender rather than trying hard to ensure that the offender’s rights are not violated. This is much more like what happens in a family where parents do what they think is best rather than trying to worry about the rights of the child.
This is in distinct conflict with the idea of due process. Under this system, the juvenile courts must be much more legalistic and formal. They have to worry more about preserving the rights of the offender instead of doing what is best for that person.