On the left and right side of the painting, Xu cleverly drew in black ink to show the moving hair on the horse's mane and tail.

解答如下：

On the left (side省略了)and right side of the painting =在这幅油画的左边和右边。

之所以使用单数可以理解为“省略side”这是最好的，也是最合理的解释。

【1】on the left side and right side of the painting 在这幅油画的左边和右边

【2】on either side of the painting 在这幅油画的任何一边

【3】on both sides of the painting 在这幅油画的左右两边

You can park on either side of the street.

=You can park on the left side and the right side of the street.

=You can park on both sides of the street.

这条街两边你都可以停车。

以上仅代表我个人观点，以上解答仅供参考。

注意区别：

The left and right sideS of the painting.

ON the left side and ON the right side of the painting.

= ON the left and right side of the painting.

如果第二句的side想用复数，可以写成：

On both sides of…

一家之言，仅供参考；勤查英英，必有所获。

两个都对，但用复数比单数的情形更多见,意思上没什么区别。请看语料库例句：

1. Asymmetry of patterns is found in left and right hands, left and right sides of brains, left and right sides of atoms, left and right sides of faces, and left and right sides of nature （复数）

2.Echocardiographic tissue characterization demonstrates differences in the left and right sides of the ventricular septum.（复数）

3.The lymphoid organ of the penaeid shrimp is located on the left and right side of the anterodorsal surface of the heptopancreas. （单数）

4.To detect mode of a component, we analyse reservoirs of the component obtained by pouring water from top, bottom, left and right side of the component.（单数）

5. It was called a 'phoney war' between the left and right wings of the British labour movement.（复数）

6. But for wider groups of parties identified with the left and right wing of the political spectrum, the picture is not much clearer. （单数）