Your "Example" statement misses an important point: CH is a sentence of set theory, and within any model of set theory CH is either true or false. There are some models of set theory in which CH is true, there are other models of set theory in which it is false, and in every model of set theory it is either true or false. Contradiction averted.
Here, perhaps, is a simpler example. Think about this sentence in group theory: $\forall a,b, \,\, a \cdot b = b \cdot a$. Would you say this sentence is either true or false? You shouldn't, but, what you might say instead is that in any model of group theory, that is, in any group, that sentence is either true or false. In words, that sentence says the group is abelian. Some groups are abelian, some aren't, and every group is either one or the other.
Your parallel axiom example is quite similar. In some models of geometry, all the axioms except the parallel axiom are true, and the parallel axiom is also true: this is the case, for example, in the Cartesian coordinate plane. But there are other models of geometry in which all the axioms except the parallel axiom are true, and the parallel axiom is false: for example, in the hyperbolic plane.
Keep in mind: "axioms" do not exist in a vacuum. We use them to study mathematical objects, sometimes called "models", and we evaluate their truth or falsity within those models. We also reason with axioms, using them to prove theorems of the form "these axioms imply those properties". From a theorem of that form, we can conclude that in any mathematical model where these axioms are true, those properties are also true.