While learning Korean, is it a good idea to start learning Chinese in parallel?
I speak Chinese, Korean and Japanese and it's not the grammar but the vocabulary which is similar. I learnt Chinese first so expected Korean to be very alike due to reading that 60% of all Korean vocab comes from Chinese. I was shocked when I found Korean so alien but in truth it is similar - to Classical Chinese and not Mandarin. I started learning Cantonese recently and indeed the vocab is far more similar to that than Mandarin. E.g. 학 / 学 which is pronounced xué in Mando but hak in both Korean and Canto.  大 is dà in Mando but dae in both Korean and Canto so University is daehak in Korean and Canto but daxue 大学 in Mando. Learning Canto made Korean make more sense to me.

The other issue that I faced was the characters. Korean uses the traditional script so upon visiting Korea I was faced with not only Korean script but a multitude of shop names which they think look cool in the traditional Chinese script. So I was struggling with two scripts! 

In general though, I've enjoyed learning them side by side as they do inevitably compliment each other, yet are different enough to not confuse you. I got to the stage of learning a new word in Mando and then looking up the word in Korean to see how similar it was. I learnt that all Chinese words in Korean work with 하다 to make them a verb so essentially all 하다 verbs are Chinese, e.g. To regret = 후회하다 / 后悔 hòu huî. It becomes quite fun!

Hope this helps! 
Gareth