Record few Americans believe in Biblical inerrancy.

[John V]

John 19:14 says it was the day before Passover.
"It was the day of Preparation of the Passover; it was about noon. "Here is your king," Pilate said to the Jews."

Mark 14:12 says on the first day of Passover
"On the first day of the Festival of Unleavened Bread, when it was customary to sacrifice the Passover lamb, Jesus' disciples asked him, "Where do you want us to go and make preparations for you to eat the Passover?"

I'll admit, I'm out of my element here, but I looked up both verses. They appear to be talking about two different days. How do you explain this?

The simplest explanation is to note that days ran from sundown to sundown. The verse in Mark is referring to the beginning of the day (i.e. evening), and the verse in John is referring to the later part of the same day. For us these would be two different days. For Hebrews, same day.

John says it was the day of preparation. Mark said they were going to make preparations.

Note that Mark 14 begins by talking about the timing:

Mark 14
1 After two days it was the Passover and the Feast of Unleavened Bread. And the chief priests and the scribes sought how they might take Him by trickery and put Him to death. 2 But they said, “Not during the feast, lest there be an uproar of the people.”

So, despite it saying a few sentences earlier that they were not going to kill Jesus during the feast, you're interpreting v12 to mean that they did kill Jesus during the feast. And note what happened after the meal that evening. Jesus sent Judas out. Jesus went out with the apostles. Men were sent to arrest Jesus. The Sanhedrin assembled and called in witnesses. All this happened at night. The problem with supposing that the meal was the Passover meal is that they were prohibited from leaving the house until the next morning. Remember, the point of Passover was that any first born not in a marked house would die.

Ex 12
21 Then Moses called for all the elders of Israel and said to them, “Pick out and take lambs for yourselves according to your families, and kill the Passover lamb. 22 And you shall take a bunch of hyssop, dip it in the blood that is in the basin, and strike the lintel and the two doorposts with the blood that is in the basin. And none of you shall go out of the door of his house until morning. 23 For the Lord will pass through to strike the Egyptians; and when He sees the blood on the lintel and on the two doorposts, the Lord will pass over the door and not allow the destroyer to come into your houses to strike you. 24 And you shall observe this thing as an ordinance for you and your sons forever.

There's no way the Pharisees are going out for any reason after the Passover meal. The John and Mark events all happened on the day (sundown to sundown) before the Passover meal.