Most scholars think King Jehoshaphat of Judah was not the (grand)father of
Jehu, and this is the reason Chronicles does not mention the fact. According to this hypothesis, Jehu's father was someone else with the same name. The Book of Kings recorded Jehu's bloodline because he became one of the kings of Israel. Chronicles downplays the importance of these kings, gives short shrift to Jehu's role. So it does not give details about his lineage.
For the author of the Book of Kings, Jehu's noble descent bolstered his claim to Israel's throne. Jehu was no mere usurper but was the son of well known family who had been appointed to his task by the prophets Elijah and Elisha. However, if we turn to the books of the prophets, we see that Hosea denounced Jehu for the massacre at Jezreel (2 Kings 10). This was the very thing that had brought him to the kingship, but Hosea declared that this bloody act was to be punished by bringing an end to his dynasty:
Hosea 1:4
Give him the name “Jezreel,” for in a little while I will punish the
house of Jehu for the bloodshed at Jezreel and bring to an end the
kingdom of the house of Israel.
The Book of Chronicles was written centuries later. It focuses on Judah and gives relatively little attention to the history of the kings of Israel. It does not mention Jehu's prophetic commission and anointing, nor does it even mention the prophet Elisha. Its author must also have been aware that Jehu's kingdom was rejected because of the means by which he came to power.
Conclusion: King Jehoshaphat was probably a different person than Jehu's father, who had the same name. In any case we may speculate that the Book of Chronicles did not draw attention to Jehu's descent because it focused on the kings of Judah and was not interested in bolstering the northern kings' claim to authority.
