I will explain the problem in a simplified version.
Three Tasks: $T_1, T_2, T_3$
Four Machines: $M_1, M_2, M_3, M_4$
The machines $M_2$ and $M_3$ make the same processing, so they are parallel.
Tasks $T_1$ and $T_3$ must go through the machines in the following order: $M_1 \to (M_2\, or\, M3) \to M_4$.
Task $T_2$ is different, not all machines need to be run through. The order is $M_1\to M_4$.
My question is now if this counts to the family of job-shop problems? Because in books I always read that in the job-shop problem every task must be processed on every machine, but the order does not matter.
If this is not a job-shop problem, to which family of the shop problems does this explanation count? It would be very helpful if you can add the book/paper where this is defined, that I can obtain more info about it?