The spring constant specifies force per compression distance.
When you put multiple spring in series, the force on each spring is the same, which is also the overall force on the combined spring. The forces don't add.
However, the displacements do add. If you have three springs in series with each compressed 50 mm, then the overall spring is compressed 150 mm.
With multiple springs in parallel, the forces add but the displacements don't.
Whenever the arrangement adds the quantity in the numerator, the constants for each element are just added. Whenever the arrangement adds the quantity in the denominator, the individual constants are combined as in the parallel resistance formula, as you show.
Whether the constants of particular elements add or are combined as parallel resistances depend on which way we define the constants, and what we consider "parallel" and "series". Both these are somewhat arbitrary choices of convention on our part. We could just as well have decided to generally think in terms of conductance instead of resistance, and define a "resistor" as the current is passes per unit voltage.