Your confusion comes from mixing three frameworks described with different mathematical models.
Classical physics where particles are massive, can be modeled with classical mechanics and classical electrodynamics ( maxwell's equations). An electron when first discovered, was called a particle with charge and assumed it has classical charged particle properties.
Classical physics could not explain a number of phenomena, particularly black body radiation, the photoelectric effect , the spectra in the light from different atoms.
This forced the invention of quantum mechanics, first the simple Bohr model, and then strict theoretical equation , the Schrodinger one to start with, and then for special relativity consistency, the Dirac, the Klein Gordon and also a quantized maxwell equation for the photons.
Simple potential problems could fit the data for two body situations, explaining the atomic spectra etc.
As experiments on these new microscopic entities progressed, two body quantum mechanical calculations were shown to be inadequate, and quantum field theory was invented, in order to model the data and predict correctly new situations.
The standard model of particle physics is mathematically a field theoretical model which at the moment describes most particle data.
In this model there are these fundamental point particles, some with mass and some with mass 0(gluon, photon)
The field theory used assumes that for each of the particles in the table each point in four dimensional space is covered by a particle field , and the particles one measures in the laboratory are excitations on these fields described by creation and annihilation operators which mathematically act on the free particle solutions of framework 2 above. To signal the existence or disappearance of an electron for example, one has to use an operator: creation operator for appearance on the electron field and a destruction operator for disappearance, on the electron field.
Now to come to your question:
Why is there not a particle associated with just the electric field or just the magnetic one?
The electric and magnetic fields are classical fields of framework 1. As nature is continuous it can be shown that the classical fields emerge from the quantum mechanical fields in a long drawn field theoretical calculation. A higher level built on the quantum underlying level. Thus light is emergent from a large number of photons in a complicated mathematically consistent manner, but a photon is not light, in the sense that a building is made of bricks , but a brick is not a building.