We have constructed color-magnitude diagrams for 10 globular clusters in the halo of the nearby spiral galaxy M33 based on Hubble Space Telescope Wide Field Planetary Camera 2 observations in the F555W (~V) and F814W (~I) filters. These data reveal the morphology of the horizontal branch (HB) and allow us to estimate the cluster metallicity using the shape and color of the red giant branch. The principal result we report herein is that eight of the 10 clusters possess exclusively red HB morphologies and yet their metallicities are as metal poor as [Fe/H]=-1.6. Indeed, these eight clusters basically present only giant branch clumps reminiscent of intermediate-age star clusters in the Magellanic Clouds. In addition, two of the clusters form a second parameter pair that have similar metallicities but very disparate HB types. Under the assumption that cluster age is the global second parameter, the average age of halo globular clusters in M33 appears to be a few gigayears younger than halo clusters in the Milky Way. Using the observed properties of HB stars in M31 and M33, along with published main-sequence turnoff ages for the globular clusters in the Milky Way, LMC, SMC, and the Sagittarius dwarf spheroidal (Sgr), we attempt to sketch the early formation history of these galaxies. This indicates that the Milky Way, M31, M33, the LMC, and Sgr all experienced their first epoch of cluster formation soon after the big bang. And 3-4 Gyr later, the SMC began to form its first generation of clusters; the bulk of the M33 clusters formed later still. We note that the halo clusters in M33 formed over a much larger time period than those in the Milky Way and M31.