Properties of Electromagnetic Waves

Electromagnetic waves are produced by accelerating charges, not by charges moving with uniform speed (a charge moving with constant velocity does not radiate). The other properties – speed $\frac{1}{\sqrt{\mu_0\varepsilon_0}}$, transverse nature and equal energy densities in the electric and magnetic fields – are all true for EM waves in free space. So 'they originate from charges moving with uniform speed' is not a property of EM waves.

Electromagnetic waves can travel through vacuum; they need no material medium (true).
$\vec{E}$ and $\vec{B}$ oscillate in phase, so they reach their maxima and minima at the same place and time (true).
The average energy densities of the electric and magnetic fields are equal: $\frac{1}{2}\varepsilon_0E^2 = \frac{B^2}{2\mu_0}$ (true).
$\vec{E}$ and $\vec{B}$ are perpendicular to each other, and both are perpendicular to the direction of propagation; they are not parallel to each other.
Hence the false statement is that the electric and magnetic field vectors are parallel to each other.