All known what animal fossils had no shells or hard preservable parts

Zygotes, algeas, and most small waterlife

In order to form fossils, organisms must have hard parts. These hard parts do not decay easily. These hard parts are quickly buried under the loose sediments like clay and silt, getting separated from the oxygenated environment and after millions of years under the required necessary conditions, it forms fossils. Examples of fossils that have hard parts are brachiopods, gastropods, and cephalopods.

Those organisms that lack these hard parts do not form fossils. Even if they form they are very rare. Some of the organisms that do not have the hard parts are corals, squids (Cephalopods), jellyfish and octopus.