You can park at the start of scenic drive and bike from there it's 6 miles to riverwalk. Then 1 more mile to the narrows start. From the south entrance it's 10 miles of biking plus all the strenuous nature of the narrows. For your 10 year old this is too much. Remember Zion is a very hot climate. To do the narrows without a stick and proper shoes is also sketchy.
From the bottom of scenic drive to the trailhead for the narrows it's 8 miles, which is doable on bike.
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