Adiabatic parcel gradient in a hydrostatic environment ($d{T}_g/dz=-(g/C_p)T_g/T_{\mathrm{env}}$)

A dry parcel in rapid pressure balance with an ambient ideal gas has $d{T}_g/dP=T_g\mathcal{R}/(P{C}_p)$. Ambient hydrostatic equilibrium gives $dP/dz=-Pg/(\mathcal{R}{T}_{\mathrm{env}})$, hence $d{T}_g/dz=-(g/C_p)T_g/T_{\mathrm{env}}$. At a temperature-matched launch point this equals the dry adiabatic lapse rate, supplying the local linear stability test. A substantially hotter parcel does not exactly follow the ambient density's hydrostatic adiabat.