Polygamy is clearly allowed under the Old Covenant it is not generally disallowed under the New Covenant. Jesus never changed the law regarding polygamy. Such an important change to the law would, surely, have been stated clearly.
The Old Testament allowed polygamy, and even gave instructions for the dividing of the inheritance: if a man have two wives. (Deut 21:15) In one case polygamy was even commanded, when, if two brothers were living together, and one brother died, the other brother was to marry his wife (Deut 25:5). This meant that if the living brother was already married, which was probably usually the case, he was to take a second wife. Had polygamy been only a concession to man's evil heart, it would, in no circumstances, have been commanded. Polygamy is sometimes God's will.
Men naturally desire to have more than one woman, although decent women only desire one man. Although the fulfillment of man's desire must be limited to what God lawfully allows and provides, his desire should not be thought sinful, because the nature of male and female were created perfectly in the garden of Eden. Although, because of weakness, Satan is able to corrupt this nature, the fundamental nature is good.
Kings of Israel were commanded not to multiply wives (Deut 17:17). This meant that although, like David, they could perhaps take eight wives (1 Chron 3:1-5), they could not, like Solomon, take seven hundred (1 Kings 11:3). Too many wives is not good, as the woman is denied sufficient time with the man. Muslims limit the number of wives to four. But it may be selfish for a man to have even this number.
That polygamy is not adultery, is evident in that the Old Covenant, which forbad adultery (Exod 21:14), but allowed polygamy. God never criticised David for taking seven of his wives. But He did criticise him for taking another man’s wife. (2 Sam 12:9)
Jesus said that invalid divorce was wrong, speaking of the permanence of the marriage bond from the beginning, and saying: they twain shall be one flesh. (Matt 19:5)
But the Old Covenant, which allowed polygamy, says: the Lord, the God of Israel saith that he hateth putting away. (Mal 2:16), and also says: did not he make one? (Mal 2:15) So Jesus was not changing the law when He spoke regarding marriage. He was clarifying it.
It is possible for a man to be united with more than one woman, just as the head is united with two eyes and two ears.
But concubinage is not good. Christ came to set free. Although the doctrine of the garden of Eden is still true, that: neither was the man created for the woman; but the woman for the man (1 Cor 11:9), the curse of servitude placed on the woman, that: thy desire shall be to thy husband and he shall rule over thee, (Gen 3:16) has been cancelled.
Now, although man still has the ultimate authority and wives should still: be in subjection to their own husbands. (1 Pet 3:1) there is more of a cooperation. The wife also has power. Paul says: the husband hath not power of his own body, but the wife. (1 Cor 7:4)
This greater freedom is not available with concubinage. For this reason, in the early church, men, in kindness, married their concubines to give them the full status of a wife.
One of the requirements of a bishop or pastor was that he must be: the husband of one wife. (1 Tim 3:2) This implies that there were some men in the church with two wives.
There are several reasons why this restriction applies to pastors only, and not to the church in general. A pastor must devote time and thought to caring for the church. If he has too many wives and children, he may take more thought for the things of the world than those of the Lord. The church also must support multiple wives and children, which may be a burden. There is also a possibility of exploitation, with every pretty young woman in the church thinking it her duty to marry the pastor. Because a pastor is to be esteemed very highly in love for his work's sake, he must be more careful than others, to teach women their freedom, along with wisdom to choose.
Deacons also were to be the husband of one wife (1 Tim 3:12). But this is because a deacon in the early church had much more power and responsibility than one who is called a deacon today. The first deacons of were responsible for the ministration of provision to a church of thousands of displaced persons, distributing from the proceeds of the sale of houses and lands by many (Acts 4:34, 6:1). They were men full of the Holy Ghost and wisdom. One of the deacons, Stephen: did great wonders and miracles among the people (Acts 6:8). Today, the limited responsibilities of one called a deacon, should not, in general, restrict his freedom to marry more than one wife. But if he has significant authority over many people, he can only take one wife.