  In this paper, we introduce and analyse an explicit formulation of fractional
powers of the parabolic Lam\'e operator $\mathbb{H}$ and we then study the
extension problem associated to such non-local operators. We also study the
various regularity properties of solutions to such an extension problem via a
transformation which reduces the extension problem for the parabolic Lam\'e
operator to another system that resembles the extension problem of the
fractional heat operator. Finally in the case when $s \geq 1/2$, by proving a
conditional doubling property for solutions to the corresponding reduced system
followed by a blowup argument, we establish a space-like strong unique
continuation result for $\mathbb{H}^s \textbf{u}=V\textbf{u}$.
