Summary: In this paper, we study the following stochastic differential equation (SDE) in \(\mathbb R^d\): \[ \text d X_t= \text d Z_t + b(t, X_t)\,\text d t, X_0=x, \] where \(Z\) is a Lévy process. We show that for a large class of Lévy processes \(Z\) and Hölder continuous drifts \(b\), the SDE above has a unique strong solution for every starting point \(x\in\mathbb R^d\). Moreover, these strong solutions form a \(C^1\)-stochastic flow. As a consequence, we show that, when \(Z\) is an \(\alpha\)-stable-type Lévy process with \(\alpha\in (0, 2)\) and \(b\) is a bounded \(\beta\)-Hölder continuous function with \(\beta\in (1- {\alpha}/{2},1)\), the SDE above has a unique strong solution. When \(\alpha \in (0, 1)\), this in particular partially solves an open problem from Priola. Moreover, we obtain a Bismut type derivative formula for \(\nabla \mathbb E_x f(X_t)\) when \(Z\) is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with Hölder continuous \(b\) and \(f\): \[ \partial_t u+\mathscr L u+b\cdot \nabla u+f=0, u(1, \cdot )=0, \] where \(\mathscr L\) is the generator of the Lévy process \(Z\).