Is it true that on a modern processor, parallelism is possible on a single core?
It is indeed possible to have parallelism on a superscalar processor. A superscalar processor can execute multiple instructions at the same time by using multiple execution units.

There are certain limitations depending on the architecture. It is not true parallelism. If you have to calculate
$$A = B + C,$$
$$D = A + 3,$$
you cannot execute both instructions at the same time. However you can execute
$$A = B + C,$$
$$D = D + 3,$$
simultaneously by utilizing two ALUs.

So as an answer to your question, you can have a certain level of parallelism on a single core, as long as your instructions do not use the same hardware resources.