Why is a stationary point of a curve given by parametric equations on a surface not a stationary point of the function itself?

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I am fine with most of this question but am unsure about the stationary point not being a stationary point of the function?

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Consider earth’s northern hemisphere; the only stationary point is the North Pole. Now let an aero plane travel from Rome to New York City. You don’t suspect that the most northern point on its flight is the North Pole.