381. ¿Es correcto este uso?: John said buying the house was the best option. On the other hand, my parents can lend us some of the money.

No. On the other hand no se puede usar simplemente para añadir un argumento, como se hace con la locución española por otro lado:

John said buying the house was the best option. On the other hand, my parents can lend us some of the money.

John said buying the house was the best option. What’s more, my parents can lend us some of the money.

On the other hand se emplea cuando se introduce un contraargumento o contraposición:

She knows the staff, which would be an advantage. On the other hand, she’s perhaps too friendly with some of them.

Paul only watches serious films – I, on the other hand, will watch almost anything.

En muchos casos se puede usar in contrast en lugar de on the other hand, aunque resulta más formal:

Her sisters have always had a desire to travel. Jane, on the other hand, never has.

Her sisters have always had a desire to travel. Jane, in contrast, never has.

A diferencia de on the other hand, se puede emplear in contrast justo delante de una noun phrase:

In contrast to/with her sisters, Jane’s never had a desire to travel.

On the one hand se puede utilizar para presentar un argumento inicial:

On the one hand, the internet means we can maintain friendships with people on the other side of the world. On the other hand, loneliness is becoming a serious social problem.

Es importante observar que on the other hand no se usa para plantear una contradicción; para eso se utiliza on the contrary:

They’re not rich! On the contrary, they’re in serious debt.

I haven’t been working hard? On the contrary, I’ve been slaving away for hours.

Básicamente, si el enunciado se puede sustituir por you’re wrong (‘estás equivocado’), hay que usar on the contrary y no on the other hand, que está más próximo a ‘por otra parte’ o ‘en cambio’.