Another common variant
子は親の背中を	2711
子は親の背を	712

子供は親の背中を	3667
子どもは親の背中を	935
こどもは親の背中を	67

背 instead of 背中 is reasonably common too (for a proverb):
子供は親の背を	448
子どもは親の背を	192

gooコロケーション辞典（和英）

a variant which gets a lot of hits on gogle:
親の背中を見て育つ 	0	
親の背中を見て子は育つ	0	


https://crd.ndl.go.jp/reference/entry/index.php?id=1000050787&page=ref_view
mentions this variant too
子供は親の後姿を           110
子どもは親の後姿を 	30

Mark Twain quote: “Children are natural mimics who act like their parents despite every effort to teach them good manners.”