In this paper, we present a general solution for time-harmonic electromagnetic fields with its electric and magnetic fields parallel to each other (E || B fields) in source-free vacuum and demonstrate that every time-harmonic E || B field is composed of the superposition of two counter-propagating Beltrami fields. We show that every E || B field can be categorized into one of two cases depending on the time dependence of the function that describes the proportionality between the electric and magnetic fields. After presenting the mathematical definition of a Beltrami field in electromagnetism and its handedness, we perform a detailed analysis of time-harmonic E || B fields for each case. For the first case, we find the general solution for the E || B fields using the angular-spectrum method and prove that every first-case E || B field can be generated by superposing two oppositely traveling Beltrami fields with the same handedness. For the second case, we deduce the general solution for the E || B fields by employing complex analysis and demonstrate that every time-harmonic E || B field is composed of two counter-propagating planar Beltrami fields with opposite handedness.