We use deep Hubble Space Telescope Advanced Camera for Surveys/High Resolution Channel observations of a field within M32 (F1) and an M31 background field (F2) to determine the star formation history (SFH) of M32 from its resolved stellar population. We find that 2-5 Gyr old stars contribute ~40% ± 17% of M32's mass, while ~55% ± 21% of M32's mass comes from stars older than 5 Gyr. The mass-weighted mean age and metallicity of M32 at F1 are langAgerang = 6.8 ± 1.5 Gyr and lang[M/H]rang = -0.01 ± 0.08 dex. The SFH additionally indicates the presence of young (<2 Gyr old), metal-poor ([M/H] ~ -0.7) stars, suggesting that blue straggler stars contribute ~2% of the mass at F1; the remaining ~3% of the mass is in young metal-rich stars. Line-strength indices computed from the SFH imply a light-weighted mean age and metallicity of 4.9 Gyr and [M/H] = -0.12 dex, and single stellar population-equivalent parameters of 2.9 ± 0.2 Gyr and [M/H] = 0.02 ± 0.01 dex at F1 (~2.7 re ). This contradicts spectroscopic studies that show a steep age gradient from M32's center to 1 re . The inferred SFH of the M31 background field F2 reveals that the majority of its stars are old, with ~95% of its mass already acquired 5-14 Gyr ago. It is composed of two dominant populations; ~30% ± 7.5% of its mass is in a 5-8 Gyr old population, and ~65% ± 9% of the mass is in an 8-14 Gyr old population. The mass-weighted mean age and metallicity of F2 are langAgerang = 9.2 ± 1.2 Gyr and lang[M/H]rang = -0.10 ± 0.10 dex, respectively. Our results suggest that the inner disk and spheroid populations of M31 are indistinguishable from those of the outer disk and spheroid. Assuming the mean age of M31's disk at F2 (~1 disk scale length) to be ~5-9 Gyr, our results agree with an inside-out disk formation scenario for M31's disk. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with GO proposal 10572.