In this paper, we study the following stochastic differential equation (SDE) in $\mathbb R^d$: $$ \mathrm d X\_t= \mathrm d Z\_t + b(t, X\_t),\mathrm d t, \quad X\_0=x, $$ 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 $f$: $$ \partial\_t u+\mathscr L u+b\cdot \nabla u+f=0,\quad u(1, \cdot )=0, $$ where $\mathscr L$ is the generator of the Lévy process $Z$.