Can magnetic force really be just electric force in a different frame? Relativity explains magnetism as the electric force observed in another frame under Lorenz transformation, is that right? electrons move in a wire in a different frame of the positive ones.
Now, even if that is somewhat stretched, it can work for a conductor, but can you specify how can that explanation make sense if 2 charges are moving in parallel direction, both at rest in each other's frame?
And, most of all how can that work for light, which is in an absolute frame of its own? If magnetic were the same as electric force/field then how can the alternate in the same frame?
 A: Yout first statement that relativity explains magnetism as the elwctric force in a different reference frame is not correct.
What is usually shown is that electric force + relativity implies a magnetic force law. But $E \cdot B$ is a relativistic invariant. (also $E^2 - B^2$ is) So if you have an electric  and magnetic field togetjer, you have this in all frames.
A: Extend your arms in front of you and consider two electrons, one moving from your shoulder to your wrist along each arm.
The electrons say:  We are standing still, and the electric force is repelling us from each other.  Therefore the left-arm electron is being forced to the left and  the right-arm electron is being forced to the right, at a rate you can calculate.
But of course, because of relativity, you see everything happening in slow motion.  The electrons are drifting to the left and right, but more slowly than you'd expect based on the electric force.  You explain this by saying there must be some countervailing force pushing these electrons toward each other.  You call that force magnetism. 
A: We know that magnetic fields are generated perpendicular to the current in a wire or a particle's motion.  So the idea does seem valid... and it makes sense, considering electricity and magnetism are really just two sides of the same coin.
As for your example of two particles moving parallel to each other, yes they would seem at rest in each of their inertial reference frames.  If this was the case however, their magnetic fields would also appear to be zero (from their reference frames).  Since each particle is not being magnetically pulled towards each other, the magnetic field by their reference is indeed zero.
I'm not entirely sure what you mean in your last example with light.  If you could clarify further I can try to provide an answer.
A: 
but can you specify how can that explanation make sense if 2 charges are moving in parallel direction, both at rest in each other's frame?

Case 1: The distance between the charges is the same in the frame of the charges and the frame of the observe that observes the charges moving: 
Special relativity tells us that we must divide the force felt by the charges by gamma, in order to get that same force in another frame. The force between charges in frame A is different than the force between those charges in frame B. When observers in frames A and B say "the force between the charges", they refer to the same force, which is different in those two frames.
Calculating the force felt by the charges is trivial in this case. Of course we need special relativity to know whether the distance is the same in both frames or not. 
Case 2: distance between charges is not the same in both frames:
Special relativity tells us we must divide the force felt by charges by gamma, to get that same force in another frame. And special relativity tells us how to calculate the force felt by the charges. The force felt by charges is inversely proportional to distance squared. And Special relativity tells us what the distance is in the frame of the charges, then we can easily calculate the force felt by the charges.
In my opinion all that above stuff is trivial, so it makes a lot of sense. 
A: It has be taken in attention that a charge has not only an electric field but also a magnetic dipole moment. 
To take it in a historical and empirical context here are some facts:


*

*The original experiments of Ampère and Gauss showing a force between two linear currents was claimed by Ampère in 1825 and Gauss in 1833.

*This time nobody cares that at the moment of the hard switch-on of the current the electrons inside the wire get accelerated and by this their magnetic dipole moments get aligned. A self-inductance takes place. This happens independent from a second current carrying wire. And for some alloys a magnetic field remains even after switching of the current.

*The electrons magnetic dipole moment which is responsible for the above point was discovered after 1924 (Source Wikipedia):  


*

*Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell.  

*In retrospect, the first direct experimental evidence of the electron spin was the Stern–Gerlach experiment of 1922. In this experiments an external magnetic field deflects moving particles which is only possible if the magnetic dipole moments of this particles is influenced.


*The coincidence between the electrons spin and the electrons magnetic dipole moment is obvious.



Can magnetic force really be just electric force in a different frame?

It's not necessary to be in a relative motion to an electric charge (in a different frame) to observe a magnetic field around this charge. The acceleration of a bundle of electrons (the current in a wire) as well as the influence of an external magnetic field led to the alignment of their magnetic dipole moments. This we observe as a macroscopic magnetic field. 
