In some ways, it is; in other ways, it's not! Just like any other dialect.
A quick example to demonstrate my point:
MSA: بإمكانك أن تحصل إما على التفاحة وإما على الموزة وليس على كلتيهما
/biʔimka:nika ʔan taħsˤala ʔimma ʕala attuffa:ħa waʔimma ʕala almawza walajsa ʕala kiltajhima:/
PA: بتقدر توخد يا التفاحة يا الموزة مش التنتين
/btiʔdar to:xod ja: ittuffa:ħa ja: ilmo:ze miʃ ittinte:n/
In Morocco, you can say
يمكن لك/تقدر تاخذ/توخذ التفاحة أو البنانة, ماهو/مو/ماشي الزوج/الاثنتين
ثنتين, ماهو/مو and توخذ are typical from the South (bedouin).
Given what's been said in this thread, that makes a lot of sense! If the origin of the word for "parsley" is the Ancient Greek word for "Macedonia," then it would make sense that the use of ب instead of م was a later development that only occurred in certain dialects, and that MSA "borrowed" the word from one of those dialects. Otherwise, we would expect م and not ب.
Indeed, MSA borrowed from dialects, or rather MSA borrowed from the use of the speakers who introduced the words into MSA. By the way, when I said Syrians, I meant Palestinians/Lebanese/Syrians all together, I was pointing at Greater Syria at the time when standardisation/modernisation occurred. So that's no surprise if words used in these dialects (Egyptian, شامي) most notably related to food, made their way into MSA while the words used in the Maghreb, Arabia, Sudan and Iraq, didn't (معدنوس may be a good example to illustrate this). For instance, how one would say "basil" if he had to write a text in MSA? Or peas? There are great chances that anyone will respectively use ريحان and بازيلاء even if these words either have a different meaning or aren't used in the person's native dialect (I mean in a broad way, no one prevents a person from using the words he/she is used to).
That's no wonder if foreigners often notice and say that Syrian/Palestinian/Lebanese/Egyptian dialects are closer to MSA when comparing the words used in these dialects with MSA. It's because many words made their way into MSA from these dialects and not from others (people often -rightfully at first- think I eat whale meat when I say حوت while I mean fish

)
French and English as well are concerned by this, not only Arabic and I think pretty much all languages who have a wide distribution face such phenomena.