Stellar and supernova nucleosynthesis in the first few billion years of the cosmic history have set the scene for early structure formation in the Universe, while little is known about their nature. Making use of stellar physical parameters measured by GALAH Data Release 3 with accurate astrometry from the Gaia EDR3, we have selected $\sim 100$ old main-sequence turn-off stars (ages $\gtrsim 12$ Gyrs) with kinematics compatible with the Milky Way stellar halo population in the Solar neighborhood. Detailed homogeneous elemental abundance estimates by GALAH DR3 are compared with supernova yield models of Pop~III (zero-metal) core-collapse supernovae (CCSNe), normal (non-zero-metal) CCSNe, and Type Ia supernovae (SN Ia) to examine which of the individual yields or their combinations best reproduce the observed elemental abundance patterns for each of the old halo stars ("OHS"). We find that the observed abundances in the OHS with [Fe/H]$>-1.5$ are best explained by contributions from both CCSNe and SN~Ia, where the fraction of SN~Ia among all the metal-enriching SNe is up to 10-20 % for stars with high [Mg/Fe] ratios and up to 20-27 % for stars with low [Mg/Fe] ratios, depending on the assumption about the relative fraction of near-Chandrasekhar-mass SNe Ia progenitors. The results suggest that, in the progenitor systems of the OHS with [Fe/H]$>-1.5$, $\sim$ 50-60% of Fe mass originated from normal CCSNe at the earliest phases of the Milky Way formation. These results provide an insight into the birth environments of the oldest stars in the Galactic halo.