Growth off Palm tree

I am not a gardener by any stretch of the imagination. We have just moved into a house with two 'palm trees' in the garden. One has another branch system growing out of one of the leaves. It's about two feet long and has many branches of its own with tiny green buds. Both trees have dead versions of these, also coming out of a leaf of the tree itself. Does anyone know what this is? Is it flowering and should I leave it? Or is it some sort of parasitic plant I need to cut off. Should I cut off the dead ones? Btw I call them palm trees but I have no idea what they really are. They are about 12 feet tall, and then have spikey leaves coming from the top. No branches. They look to me like big versions of the plants everyone used to get from ikea 15 years ago.