Essentially are very close in meaning.
For a native (prone to the mistake above) "debe de" implies a stronger supposition, or even a "must" for your dad, to be proud. But strictly speaking is that he probably is proud.
The one with "debería de", it is a supposition but also can have the idea that "if your dad is not proud", he "should be proud", it would be something expected, whether it is the case or not.
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Lo propio de la verdad es que se basta a sí misma, aquel que la posee no intenta convencer a nadie.
"An enemy is somebody who flatters you. A friend is somebody who criticizes the living daylights out of you."
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