Think that you have two Riemannian metrics in a neighborhood of the origin in $\mathrm{T}_p$, the first is standard Euclidean and the second is induced by $\mathrm{exp}_p$ from $M$.

These two metrics coincide at the origin up to second order. It is sufficient to conclude that principle curvatures at the origin calculated in both metrics are the same.

Think that you have two Riemannian metrics in a neighborhood of the origin in $\mathrm{T}_p$, the first is standard Euclidean and the second is induced by $\mathrm{exp}_p$ from $M$.

These two metrics coincide at the origin up to first order. It is sufficient to conclude that principle curvatures at the origin calculated in both metrics are the same.