Yes, electromagnetic waves can have both parallel and perpendicular components of the electric field (E-field) and magnetic field (H-field) simultaneously. This property is known as polarization.
The polarization of an electromagnetic wave describes the orientation of the electric and magnetic fields with respect to the direction of wave propagation. An electromagnetic wave can be linearly polarized, circularly polarized, or elliptically polarized.
For linearly polarized waves, the electric and magnetic fields oscillate in a single plane perpendicular to the direction of propagation. In this case, the E-field and H-field can have components both parallel and perpendicular to the direction of wave propagation.
When an electromagnetic wave is circularly polarized, the E-field vectors rotate in a circular manner as the wave propagates. This means that the E-field and H-field have both parallel and perpendicular components, and the orientations of the fields change as the wave propagates.
Elliptically polarized waves are a combination of linear and circular polarization. The E-field and H-field vectors trace out an elliptical pattern as the wave propagates. Again, this means that both parallel and perpendicular components of the fields exist.
In summary, electromagnetic waves can have both parallel and perpendicular components of the electric and magnetic fields, and the polarization of the wave determines the specific nature of these components.