We are wondering what the "Yeast of the Pharisee's" is and what part it has in the story?

Your question deals with “the yeast of the Pharisees.” Please do not take what I am about to say as a criticism of you or your question. It is not. It is a criticism of the new versions. Many people who use the new versions are good people who have been told that the new versions will help them understand the Bible better. Their motives are pure, so I have no axe to grind with them.

First of all, the “yeast of the Pharisees” is a phrase used in many of the “new” versions to replace what the King James Bible calls, “the leaven of the Pharisees.” Leaven is a picture of sin in the Bible. 1 Corinthians 5:6-8 [6] Your glorying is not good. Know ye not that a little leaven leaveneth the whole lump? [7] Purge out therefore the old leaven, that ye may be a new lump, as ye are unleavened. For even Christ our passover is sacrificed for us: [8] Therefore let us keep the feast, not with old leaven, neither with the leaven of malice and wickedness; but with the unleavened bread of sincerity and truth.
See All... says, “Your glorying is not good. Know ye not that a little leaven leaveneth the whole lump? Purge out therefore the old leaven, that ye may be a new lump, as ye are unleavened. For even Christ our passover is sacrificed for us: Therefore let us keep the feast, not with old leaven, neither with the leaven of malice and wickedness; but with the unleavened bread of sincerity and truth.

There can be no doubt that yeast is a leavening agent, but it certainly is not the only one. Most kitchens contain baking soda and baking powder which are also leavening agents. The point is that leaven is a broader term that encompasses all leavening agents. Yeast only covers one type. The Bible defines sin in the broadest of terms in 1 John 5:17All unrighteousness is sin: and there is a sin not unto death.
See All... (“All unrighteousness is sin…”). So leaven makes an effective picture of sin because both terms are broad. Now this is important because there is more than one answer to your question. In Matthew 16:12Then understood they how that he bade them not beware of the leaven of bread, but of the doctrine of the Pharisees and of the Sadducees.
See All..., leaven is defined as “the doctrine of the Pharisees and Sadducees.” That is, it refers to the false teachings of these two religious groups. Although they were teaching some good things, they also had some false doctrine mixed in. Again we are reminded of 1 Corinthians 5:6Your glorying is not good. Know ye not that a little leaven leaveneth the whole lump?
See All...—“…a little leaven leaveneth the whole lump.” In Luke 12:1In the mean time, when there were gathered together an innumerable multitude of people, insomuch that they trode one upon another, he began to say unto his disciples first of all, Beware ye of the leaven of the Pharisees, which is hypocrisy.
See All... where the leaven of the Pharisees is again mentioned, it is defined as “hypocrisy.”

Second, one of the supposed benefits of the new versions is that folks are better able to understand the Bible because it is in modern English instead of the archaic Elizabethan English of the King James Bible. Changing leaven to yeast didn’t clear anything up. The answer is found by studying the word of God. Of course, the new versions remove the word study from their bibles (2 Timothy 2:15Study to shew thyself approved unto God, a workman that needeth not to be ashamed, rightly dividing the word of truth.
See All...) so that their readers won’t feel compelled to search the scriptures.