Churches burn the previous year's blessed palm fronds for the current year's Ash Wednesday blessed ashes. You might check with your parish to find out if and when the staff will be collecting palm fronds from the parishioners. If your parish is not collecting palm fronds, you can dispose of the palm fronds by burning them and return the ashes to the ground (perhaps by spreading them in a garden).
__________________ "If anyone comes to me, I want to lead them to Him." --St. Edith Stein