You could rewrite 'the fathers and mothers' as 'the fathers and the mothers', but you can't write 'both Jack and both Jane'.
You seem to be trying to compare apples with oranges :)
"Both...and..." is a linking phrase composed of the two conjunctions the first one of which explicitly incorporates the idea two or more following things that all are included.
So, using the extra "both" after "and" would be neither reasonable nor logical, to say the least.