"I am already taller than my father."
Translation:Už jsem vyšší než můj otec.
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Because you can view it as shortened "než je můj otec", so the subject is the father, not you. Notice that otec is in the nominative case here.
"svůj otec" in nominative is really strange because it is generally used as a subject or with "být" and "Je svůj otec." would mean "He is his own father.".