Would his parents understand him the way you do?
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Would his parents understand him how you do?
is the rephrasing above grammatically ok?
Your phrase is grammatical. The adverb, 'how', means
In what manner or way; by what means ...
(how. (n.d.) American Heritage® Dictionary of the English Language, Fifth Edition. (2011). Retrieved September 11 2015 from http://www.thefreedictionary.com/how).