It should be true for $p>1$. For a function $\varphi$ in the Schwarz class it holds that \begin{equation} D_1D_2 \varphi(x) = -R_1 R_2 \Delta \varphi(x), \end{equation} where $R_1, R_2$ are the Riesz transforms (See Grafakos, Classical Fourier analysis, Proposition 5.1.17). Therefore \begin{equation} \Vert D_1 D_2 \varphi \Vert_p = \Vert R_1 R_2 \Delta \varphi \Vert_p \leq C_{p} \Vert \Delta \varphi \Vert_p \leq C_{p} (\Vert D_1^2 \varphi \Vert_p + \Vert D_2^2 \varphi \Vert_p), \end{equation} by the $L^p$ boundedness of the Riesz transforms. For $p=1$ maybe one should look for a counterexample.

It should be true for $p>1$. For a function $\varphi$ in the Schwartz class it holds that \begin{equation} D_1D_2 \varphi(x) = -R_1 R_2 \Delta \varphi(x), \end{equation} where $R_1, R_2$ are the Riesz transforms (See Grafakos, Classical Fourier analysis, Proposition 5.1.17). Therefore \begin{equation} \Vert D_1 D_2 \varphi \Vert_p = \Vert R_1 R_2 \Delta \varphi \Vert_p \leq C_{p} \Vert \Delta \varphi \Vert_p \leq C_{p} (\Vert D_1^2 \varphi \Vert_p + \Vert D_2^2 \varphi \Vert_p), \end{equation} by the $L^p$ boundedness of the Riesz transforms. For $p=1$ maybe one should look for a counterexample.