Are components or combinations of components with common currents, in a series–parallel circuit, in either series or parallel with each other?
Let's think of Kirchkoff's first law: at any junction, the currents coming in add up to the currents going out.
If two components are in parallel, they have to have the same sum of currents. So they could split the current any way they want.
If two components are in series, they must have the same currents because you can imagine that the junctions only have one path, so all the same current goes between them.
So components with common currents are in series.
For bonus intuition, if we want to flip the question around and ask what is conserved with components in parallel, we can use Kirckhoff's second law: the Loop law! This says that any path around a circuit has a net 0 voltage jump. Therefore, any two components in parallel must have the same voltage drop because you could go either way to get to the end of a loop.