We analyze the seismicity rate immediately after the 2004 Mw6.0 Parkfield, California, earthquake from near-source seismograms. By scrutinizing high-frequency signals, we can distinguish mainshock coda from early aftershocks occurring as soon as 30 s after the mainshock. We find, as expected, that a significant fraction of aftershocks in the first few hours after the main shock are missing in the Northern California Seismic Network catalog. We observe a steady rate of aftershocks in the first 130 s, followed by a power-law decay of aftershock activity. Thus, there appears to be a distinct early stage of aftershock activity that does not fit the Omori's law with a constant p value, a phenomenon that we refer to as Early Aftershock Deficiency (EAD). Our observation suggests that mainshock rupture and aftershocks are distinct processes, not described by a single Omori's law. Several physical models of aftershocks can explain the EAD.