The confusion here is reminiscent of the discussion on Neal Whitman's Literal-Minded blog about the expression, "I need to pee like a racehorse." Neal explains, "This is an example of an attachment ambiguity, in that we could theoretically attach the modifier like a racehorse to the 'lower' verb phrase pee or to the 'higher' verb phrase need to pee." Thus, it's not:
I [ need to pee ] [ like a racehorse ]
I [ need to [ pee like a racehorse ] ]
In the "Frazz" strip, the ambiguity works the other way. The teacher thinks like my dad attaches to the lower verb be:
I [ want to [ be the guitarist for Iggy and the Stooges like my dad ] ]
when what's intended is an attachment to the higher verb want:
I [ want to be the guitarist for Iggy and the Stooges ] [ like my dad ]
See Neal's blog for more examples of attachment ambiguity, many helpfully parsed with syntactic tree diagrams. And see also Arnold Zwicky on "the lure of Low Attachment" here (with links back to relevant LL posts).