I am a first time buyer, bought a property last year and will hopefully start building this year.

If not I wouldn't bother selling as there is an incredible amount of money lost in buying and selling properties, so you have to make sure you have a plan to maximize the equity and actually have enough. 

A buyer has approached his realtor and explained that he (the buyer) would like to buy the property as an investment to rent but that he can only make money on renting it if he assumes my brother's mortgage.

Are we talking posh along "lake formerly known as Calhoun" property or an off Broadway "strap on the flak jacket" sec8 property? 

Multiple offers situations are the rare exception, not the norm.

Personally, I would focus more on the long-term cash flow of the property rather than worry about your parents living standard.

I just did a cash out refi in order to move my 1st property from my name to my LLC another cash out refi on the same property wouldn't be over kill?

I have one property my wife owns, and soon we will have two.

We do not lend in Missouri however if the property in is MO. 

Rehabbed entire property inside and out for 140k.