I want to transfer money to my parents in India. What would be the best way to transfer money (send money to NRE account or send money to my parent's account)?
Product or Service recommendation is offtopic. You can transfer money to your own NRE account or parents account. There are quite a few services.
My parents would use that money for buying shares and property on my name, and occasionally sell those shares (Would the profit which will be earned by selling those shares taxable in USA or India)?
If the intent is to repatriate the funds back to US; it is advised that you transfer the funds to your NRE account and give power of attorney to your parents. This will make it easy to repatriate funds.
From an Indian tax point of view; any gains will be taxable in India as long term gains when redeemed. Note there is a separate DEMAT - PINS account that need to be opened in your name.
From US tax point of view; ensure the all Indian Accounts / Assets are declared appropriately. Gains will be taxed in US as well. You can claim benefit under Dual Tax Avoidance Agreement. There are quite a few questions covering this subject. Search under DTAA tag
If I send money to my parents' account, can they use that money for their business and what would be tax implications on the money they earn in the business through the money I sent to them?
You can gift unlimited amount to your parents without any tax implication to them. Any interest / gains / profits from business will be taxable to your parents and they have to pay tax on gains. If the amounts are large, please document a gift deed so that paperwork is clear.