I know that Solomon was a king, but why doesn't the Bible describe his polygamy as sinful? Was he allowed to commit sins others could not because he was a king? I thought God showed no partiality.
Although the narrator in Kings and Chronicles may not say so, Deuteronomy does say it was a sin, in Deuteronomy 17:15-17
 you shall surely set a king over you whom the Lord your God chooses; one from among your brethren you shall set as king over you; you may not set a foreigner over you, who is not your brother.  But he shall not multiply horses for himself, nor cause the people to return to Egypt to multiply horses, for the Lord has said to you, ‘You shall not return that way again.’  Neither shall he multiply wives for himself, lest his heart turn away; nor shall he greatly multiply silver and gold for himself. (NKJV)
Solomon broke all these provisions, multiplying horses, wives, and gold and silver.
All King Solomon’s drinking vessels were gold, and all the vessels of the House of the Forest of Lebanon were pure gold. Not one was silver, for this was accounted as nothing in the days of Solomon. (1 Kings 10:21 NKJV)
Accounted as nothing because he had so much of it.
The Bible never says Solomon's multiple wives was not a sin. It was actually the reason he lost his kingdom. In 1 Kings 11:4 his multiple wives drew him away from his full devotion to The Lord and eventually to other gods. In 1 Kings 11:11, The Lord tells Solomon that because he did not keep God's covenant, He would take away Solomon's kingdom and give it to one of his subordinates - which ended up being his son. We are told this is for the sake of David in v. 12.
The simple and quick answer is because plural marriage (i.e. polygamy) isn't a sin according to Scripture. Solomon was instructed not to take foreign wives because they would lead his heart away to other gods. THAT was the sin that Solomon committed. It's not a sin for a man to be married to more than one woman.