  Let $v(x)$ be the numerical radius of an element $x$ in a $C^*$-algebra
$\mathfrak{A}$. First, we prove several numerical radius inequalities in
$\mathfrak{A}$. Particularly, we present a refinement of the triangle
inequality for the numerical radius in $C^*$-algebras. In addition, we show
that if $x\in\mathfrak{A}$, then $v(x) = \frac{1}{2}\|x\|$ if and only if
$\|x\| = \|\mbox{Re}(e^{i\theta}x)\| + \|\mbox{Im}(e^{i\theta}x)\|$ for all
$\theta \in \mathbb{R}$. Among other things, we introduce a new type of
parallelism in $C^*$-algebras based on numerical radius. More precisely, we
consider elements $x$ and $y$ of $\mathfrak{A}$ which satisfy $v(x + \lambda x)
= v(x) + v(y)$ for some complex unit $\lambda$. We show that this relation can
be characterized in terms of pure states acting on $\mathfrak{A}$.
