Parallel axes between inertial frames in Special Relativity In "CLASSICAL ELECTRODYNAMICS" by J.D.Jackson, 3rd Edition , $\S$ 11.3, the
 author gives in equation (11.19) a generalization of  Lorentz transformation as follows :  
If the axes in K and K' remain parallel, but the velocity $\:\mathbf{v}\:$ of the frame
K' in frame K is in an arbitrary direction, the generalization of (11.16) is 
$$
\begin{align}
x'_{0} & =\gamma\left(x_{0}-\boldsymbol{\beta}\boldsymbol{\cdot}\mathbf{x}\right)\\
\mathbf{x}^{\prime} & = \mathbf{x} +\dfrac{\left(\gamma-1\right)}{\beta^{2}}\left(\boldsymbol{\beta}\boldsymbol{\cdot}\mathbf{x} \right)\boldsymbol{\beta}-\gamma\boldsymbol{\beta}x_{0}
\end{align}
\Biggr\}
\tag{11.19}
$$
where
$$
\begin{align}
\boldsymbol{\beta} & = \dfrac{\mathbf{v}}{c}\; \qquad  \beta=|\boldsymbol{\beta}| \\
\gamma &=\left(1-\beta^2 \right)^{-1/2}
\end{align}
\tag{11.17}
$$
and
$$
\begin{align}
x'_{0} & =\gamma\left(x_{0}-\beta x_{1}\right)\\
x'_{1} & =\gamma\left(x_{1}-\beta x_{0}\right)\\
x'_{2} & =x_{2}\\
x'_{3} & =x_{3}
\end{align}
\Biggr\}
\tag{11.16}
$$
the Lorentz Transformation with the velocity $\:\mathbf{v}\:$ parallel to the common $\:x-x'\:$ axis.  
In case (11-16) it's permissible to talk about parallel axes. But in the generalized case (11-19) is it permissible to talk about parallel axes ? What is the meaning of parallelism in this later case ?
 A: In general, Lorentz (or rather Poincare) transformation are those transformations that keep the speed of light in any reference frame the same. They can be decomposed into the following:
1.) Translations 
2.) Rotations and 
3.) Boosts 
Translations and Rotations are defined just like in the Galileo case and don't make up the "interesting" physics of special relativity. Therefore, one often only really talks about the boosts when talking about Lorentz transformations. These boosts can be done in some coordinate direction, most easily in the direction of an coordinate axis, for example the $x$-axes. The corresponding transformation is written in your equation (11.16). Now Jackson talks about a generalization, by that he means here to write down the transformation for a boost in $\overrightarrow{v}$-direction which he does in (11.19). In principle it's the same kind of transformation. So in the same fashion as for the boost in $x_1$ direction, the spatial coordinate axes are parallel, i.e. the transformation does not include any rotation. If you were to shoot an arrow in $\overrightarrow{x}_i$-direction, $i=1,2,3$, the observer in the reference frame with primed coordinates would also observe the arrow going in $\overrightarrow{x}'_i$-direction.  
The parallelity refers to the spatial coordinates only.  
