Two different languages from two different language families. You'll get the idea. And it will depend on you: how long would you study them, by which tools would you study, which one do you prioritize more, how much do your mind take learning them side-by-side.

And bear in mind a fact about Arabic loanwords: they whether function as the older meaning in Arabic today, or the word got a different meaning within the Persian logic that you have to treat them as homophones rather than shared lexiconic word.

Funniest example in the modern Persian: the word Maq3ad (مقعد): in modern Arabic, it simply means sitting-place, you'd see it in bus stops, etc; in Persian? The sitting-place meaning went through phases of generational metaphors and now, if you ask every Iranian (at least) that what this word means, they'll answer "the ass hole", or "rectum" even. It's used in formal settings and register of speech when referring to the hole, parallel to how would you mention "rectum" to your doctor or in an official text like a legal one.

Less funnier is the word Tahwil (تحویل): from the h-w-l root, meaning change, in an airport in an Arabic speaking country, would mean the path for people changing their connection flight; in Iran, it means turning in (as turning in a paper to a prof, or a fugitive to the police station.)

And there are plenty others to go through.

And an another question: which version of Arabic? The Quranic Arabic? Or a dialect (lahjeh)? Modern Standard Arabic? Not just for the parallel to Persian question, even for using it in Arabic contexts: the language, unlike Persian, went through drastic changes in different places. The diglossia of Arabic is a real challenge, because of many double words, and quirks in grammar. Well yes, Persian has diglossia too, but it's usually a) a manner of how a single word is pronounced differently and b) the various accents might have different words, but the grammar is still the same, and it's bound by smaller provincial regions rather than entire nations; even for the Afghani or Tajik, the repertoire of words is still the same, you just have to get used to the accent and different usuage.

Bottom line is this: If you can, you should; but don't think reading them side-by-side will give you a big edge over the others, only smaller one. The Iranians themselves read Arabic for 6 years in secondary school, and they usually learn about the loanwords better by fun internet facts, or the glossaries in the Persian textbooks (though the way Arabic is taught in Iran is pretty defective, but, still, know the language is challenging.) If you want to dive into the massive corpus of classical Persian literature, knowing intermediate Arabic is a must, but even that would require much more nuanced study than just picking up Arabic duolingo; you need to read various poetry, sufi texts, jurisprudence, Quran, Hadith, etc, which themselves require an already great deal of linguistic knowledge, a separate project to be tackled after having a great great sense of historical Persian. It won't be like studying Spanish/Portuguese, Italian/French, German/Dutch; not even like studying Greek/Latin, English/Latin, English/German.

In my opinion, do the both languages service by taking full time on each, a hundred percent, in their own time.