The period of time during which an "explosion" of fossilized animals with bilateral symmetry are first found is the:

The Cambrian. The Cambrian Explosion, which began roughly 545 million years ago, was a relatively rapid (~5-10 million years) diversification of multicellular life, including most of the animal phyla currently extant today. This was an evolutionary arms race of sorts, with many creatures achieving both larger size as well as symmetrical bodies and various structural supports, including skeletal systems. It was also a highly experimental times in terms of evolution, as many forms of creatures arose and went extinct in rapid succession during this period.