In Lent, we move closer to holy week, a tradition that has been adapted from the Jewish feast of Passover. Their feast reminds us of the Exodus from Egypt to deliverance at the Red Sea. Jesus entered Jerusalem to join the Passover celelbration; and crowds hailed, "Hosanna! Blessed is he who comes in the name of the Lord."
But what do palm branches have to do with Passover? Moses did not give instructions to "take your palm branches with you." The Israelites did not wave them at the drowning Egyptians.
They were significant, but I contend they had very little do with the Passover. People used them quite often for the old Jewish festival of sukkot, or what we call the Festival of Booths. When the Israelites arrived at Sinia, people constructed booths made of wood. The roofs were made from freshly cut willow branches or palm branches that would have grown naturally in the desert oases. Their festival tradition was to go out, live in tents, and like the Israelites who made these tents and then built them at the base the mountain when God gave them a law to purify their lives. Out in the wilderness, the children would sing a song that reminded them that God would protect them. It was the song “Ho-shanna,” God save us. The children put a little celebration twist on it because for them, they believed that God would do it and had saved by providing the law. Despite being in the desert, they were ready to sing to God knowing ahead of time that he saved them.
(FMI on sukkot see Stephen Wylen, The Jews in the Time of Jesus: an Introduction, 101)