We embed the rotation group, $SO(3)$ into the Lorentz group, $O(1,3)$ : $SO(3) \hookrightarrow O(1,3)$ and then determine the six generators of Lorentz group: $J_x, J_y, J_z, K_x, K_y, K_z$ from the rotation and boost matrices.
From the number of the generators we realize that $O(1,3)$ is a six parameter matrix Lie group.
But are there any other way to know the number of parameters of the Lorentz group in the first place?
From special relativity we know that a Lorentz transformation: \begin{equation} x'^\mu = \Lambda^\mu {}_\nu x^\nu \end{equation} preserves the distance: \begin{equation} g^{\mu \nu} \Delta x_\mu \Delta x_\nu = g^{\mu \nu} \Delta x_\mu' \Delta x_\nu' \end{equation} The above two equations imply: \begin{equation} g^{\mu \nu} = g^{\rho \sigma}\Lambda_\rho {}^\mu \Lambda_\sigma {}^\nu \end{equation} Now, let us consider an infinitesimal transformation: \begin{equation} \Lambda_\nu {}^\mu = \delta_\nu{}^\mu + \omega_\nu{}^\mu + O(\omega^2) \end{equation} such that we can write: \begin{equation} \begin{aligned} g^{\mu \nu} & = g^{\rho \sigma}\Lambda_\rho {}^\mu \Lambda_\sigma {}^\nu \\& = g^{\rho \sigma} \left( \delta_\rho{}^\mu + \omega_\rho{}^\mu + \cdots \right)\left( \delta_\sigma{}^\nu + \omega_\sigma{}^\nu + \cdots \right) \\& = g^{\mu \nu} + g^{\mu \sigma} \omega_\sigma{}^\nu + g^{\rho \nu} \omega_\rho{}^\mu + O(\omega^2) \\& = g^{\mu \nu} + \omega^{\mu\nu} + \omega^{\nu \mu} + O(\omega^2) \end{aligned} \end{equation} and so: \begin{equation} \omega^{\mu\nu} = - \omega^{\nu \mu} \end{equation} Thus, the matrix $\omega$ is a $4 \times 4$ antisymmetric matrix, which corresponds to $6$ independent parameters (i.e. the $3$ parameters corresponding to boosts and the $3$ parameters corresponding to rotations).
It's the same way you know there are three parameters in $SO(3)$. The equation $\Lambda^T \eta \, \Lambda = \eta$ has $(n^2+n)/2$ independent scalar equations. To see this, write the equation in component form: $\Lambda^{\mu\nu} \Lambda_\mu{}^\rho = \eta^{\nu\rho}$. Now we see there are $n^2$ scalar equations equations, but because $\eta$ is symmetric and the left hand side is symmetric in $\nu$ and $\rho$ as well, the equations related by switching $\nu$ and $\rho$ are the same. Thus we have established that there are $(n^2+n)/2$ independent scalar equations.
Since $\Lambda$ has $n^2$ components, we get $n^2-(n^2+n)/2 = n(n-1)/2$ degrees of freedom. In $3D$ this comes out to $3\cdot 2 /2=3$, and in $4D$ this comes out to $4 \cdot 3/2=6$.
You've got two very good answers from Hunter and NowIGetToLearnWhatAHeadIs. However, it's probably useful to know that this beast $O(1,3)$ is isomorphic or locally isomorphic (i.e. has the same Lie algebra) to a surprising number of other interesting groups, which each give you a slightly different way to think about it. First note that its identity connected component $SO^+(1,3)$ of orthochronous, proper Lorentz transformations (those that keep the orientation of space and time the same, also called the "restricted" Lorentz group) of course determines the Lie algebra.
$SO^+(1,3)\cong {\rm Aut}(\hat{\mathbb{C}}) \cong PSL(2,\mathbb{C})$ is isomorphic to the Möbius group of all Möbius transformations, in turn isomorphic to the group of all conformal transformations of the unit sphere. So it is defined by $z\mapsto \frac{a\,z+b}{c\,z+d}$ with $a,\,b,\,c,\,d\in\mathbb{C}$ and $a\,d-b\,c=1$. So there are three independent complex parameters, i.e. six independent real parameters;
The double cover of $PSL(2,\mathbb{C})$, namely $SL(2,\mathbb{C})$ (still locally isomorphic to $SO^+(1,3)$) is the group of all $2\times 2$ matrices of the form:
$$\exp\left(\frac{1}{2}\left[\left(\eta^1 + i\theta \gamma^1\right) \sigma_1 + \left(\eta^2 + i\theta \gamma^2\right) \sigma_2 + \left(\eta^3 + i\theta \gamma^3\right) \sigma_3\right]\right)$$
where $\sigma_j$ are the Pauli spin matrices, $\theta$ is the angle of rotation, $\gamma^1,\,\gamma^2,\,\gamma^3$ are the direction cosines of the rotation axis and $\eta^1,\,\eta^2,\,\eta^3$ the components of the rapidities of the Lorentz transformation. So it's just like the general matrix $\exp\left(\frac{\theta}{2}\left(\gamma^1 \sigma_1 + \gamma^2 \sigma_2 + \gamma^3 \sigma_3\right)\right)$ in $SU(2)$ but with three complex parameters, rather than three real ones ($\theta \gamma^j$) for $SU(2)$. So again we see six real parameters.
