How to Interpret John 19:14a and John 13:1a?

Mark’s Gospel says it was the third hour (9 am according to Jewish time) when Christ was crucified (Mark 15:25). John’s Gospel mentions the sixth hour
(12 am according to Jewish time) (John 19:14a).

Both Gospel writers are correct in their claims. Each Gospel writer uses different
time systems: Mark’s Gospel follows the Jewish time system while John
follows the Roman time system. The Jewish 24-hour period began in the evening at 6 pm and the morning of that day began at 6 am. According to Roman time,
the day extended from midnight to midnight as it does for us in our time.

Mark 15:25 clearly states that Christ was crucified at the third hour Jewish time which was around 9 am. "And it was the third hour when they crucified him."

John 19:14a says: "Now it was the day of Preparation of the Passover. It was about the sixth hour." There are two possible interpretations of this:

• This occurred at 6 pm according to Roman time: 
  The preparation of the Passover Lamb occurred in the evening just after sunset (6 pm) at the beginning of the day of Passover, see Preparation of the Passover Lamb at Jesus' Last Days before his Crucifixion. 
• This occurred at 6 am according to Roman time: 
  John describes the trial before Pilate as occurring around 6 o'clock in the morning. See this further described in Jesus before Pilate (19:13-14)
  in John's Gospel: Counting Time.

Using Roman time is consistent with other references to time in John’s Gospel: 

John tells us that Jesus was tired from his journey from Judea to Samaria at the sixth hour (John 4:6) and asked the woman at the well for water. Given the length of his journey, his fatigue, and the normal evening time when people would come to the well to drink and water their animals, this fits better with 6 pm according to Roman time than 12 am according to Jewish time. See this further described in
The Samaritan Woman (4:6) in John's Gospel: Counting Time.

John's reference to the tenth hour (John 1:39) would be 10 am according to Roman time instead of 4 pm according to Jewish time. See this further described in The First Disciples (1:35-39) in John's Gospel: Counting Time.

John's reference to the seventh hour (John 4:52) would be 7 pm according to Roman time instead of 1 pm according to Jewish time. See this further described in The Official at Capernaum (4:46-54) in John's Gospel: Counting Time.

John 13:1a ("Now before the Feast of the Passover") was before Passover
according to Roman time but actually at Passover according to Jewish time, see Preparation of the Passover Lamb at Jesus' Last Days before his Crucifixion.

An alternative explanation is that the time mentioned in John 19:14a was a copying error. Greek letters were also used to write numbers. An early copyist may have confused the letter for three with the letter for six. However, there is no manuscript evidence for this suggestion, making it less likely, and therefore dismissed.

We can't know why John, as a Jew, accepted in his Gospel a form of counting days and hours that came outside his normal culture and way of counting. But we can safely say that accepting the Roman way of counting time in his Gospel provides an adequate and better understanding of certain biblical accounts;
it provides a good explanation of what are otherwise troubling differences between John’s Gospel and the Synoptic Gospels (Matthew, Mark, and Luke).

See also Critical verses in the gospels at the end of The Four Gospels.