according to the persisting theories (theories to satisfy various behaviour of light in several events) it depends on which model of light you are taking to satisfy the any specific property of light successfully. so wave models dosen't simply require the need of any parameter of mass to rely upon for thier existence, mathematically they could exists on papers without having mass. however, giving mass to wave would defy wave theory in simpe model. And other models of light they simply require the mass parameter as u may be konwing.
In general saying as these are individual therories these kind oddities could happen,but combining these theories to get upon a fundamental model may just help solve the problem.(which has not happend yet).infact many physicists have turned to a more unique physcial model to define everything- string theory.
at last who knows may scientist drop the parameter of mass and find a more richful than this to satisfy all its properties.