Islam rose at a time when Byzantium and Persia were at their weakest. Was this mere coincidence or were the destructive wars of Byzantine and Persia indirectly helping in the Rise of Islam or the unity of the Arabs?

There can be no reasonable doubting that the weakness of the Byzantine and Sassanian Empires was a critical factor in the early expansion of Islam. The critical thing to note is that early Islam was political as much as religious: when you look at maps like this one documenting the “Rise of Islam” they are referring not to the spread of a message by evangelical proselytisers, but to the growth of an empire that had been founded by Muhammad and continued after his death. (Of course, these terms 'political' and 'religious' are from the modern English language, it is far from clear – to say the very least – that seventh-century observers would have recognised such a distinction). This empire spread as virtually all empires do: by winning battles against its enemies and then claiming lands of the defeated as conquests. According to the traditional narrative, based primarily on the account of the tenth-century Persian historian Al-Tabari, the two most important such battles (amongst many) were the Battle of Yarmouk against the Byzantines and the Battle of al-Qadisiyyah against the Sassanians, both in 636. Once we note what is meant by a phrase like “Rise of Islam” the answer to your question becomes painfully obvious: of course the weakness of the Byzantines and Sassanians helped the Muslims, for the insultingly simple reason that armies are easier to defeat when they are weak than when they are strong.

In fact, we can even go slightly further than this in two ways. First, not only does seventh-century “Rise of Islam” in formerly Byzantine and Sassanian territories refer to the expansion of a political regime, it absolutely does not refer to the religious conversion of the people of these territories. We of course do not have accurate – or any – census data from the seventh-century, but to shrug one's shoulders and admit that our curiosity cannot really be satisfied is not the historian's way. Enter Richard Bulliet, who proposed a particularly interesting methodology: take the adoption of Arabic names as a proxy for conversion, counting backwards from later names which record the line of descent back to the first Arabised name. Bulliet found a great degree of regional variation – conversion was much faster in Iran than Egypt, for instance – but in all cases it would be hundreds of years before Muslims made up the majority. Summarising the reception to his work in a later lecture (link), Bulliet claims that everyone liked his rough timelines for conversion, but no one liked his methodology. This summary of course reflects his own disgruntlement; perhaps a better summary would be that of Hugh Kennedy, who states that Bulliet's figures are widely recognised as conjectural, but that they're the best we can get (Kennedy 2016, pp. 173-174). At any rate, it is widely agreed that the religion had few early converts from the initial populations. Second, attentive readers will have noted my use of the phrase “traditional narrative” when referring the Battles of Yarmouk and al-Qadisiyyah: this hedge references the fact that for early Islam, as for so many long accepted narratives about pre-modern history, a school of thought has arisen that takes a much more sceptical approach. The founding text of this school is 1977's Hagarism, authored by Michael Cook and Patricia Crone. Cook and Crone made standard sceptical moves against the traditional texts – they're late, biased, contradict earlier sources etc. – and instead proposed a radical revision where Muhammad's movement was essentially a Jewish one known as 'Hagarism' (a reference to Hagar, mother of the biblical Ishmael) rather than 'Islam' (meaning submission, in this case to God) that sought to conquer Jerusalemand eventually ended up conquering much more. The key thing to note here is that even they, the founders of the hyper-sceptical school, didn't doubt the expansion of an empire via conquest in the mid seventh-century. (Assigning names and dates to particular battles and leaders is another matter). So the answer to your question is one of the things that would be most uncontested and we can know most firmly about early Islamic history.

TLDR: yes, it definitely helped. That's just how war works. If one talks about a “Rise of Islam” as contemporaneous with Byzantine and Sassanian weakness then it's military success we're talking about, and the significance of these military successes is one of the things we can be most certain of for this period.

Sources

Richard Bulliet. 'History of the World to 1500 CE', Lecture 10.

Summarises Bulliet's work into early Islamic conversion and compares it to conversion for early Christianity and Buddhism. Provides insights into his mind that are arguably lacking in the more polished proper publication.

Richard Bulliet, Conversion to Islam in the Medieval Period, 1979.

The more polished proper publication. Goes into much greater detail, gives actual numbers and discusses regional variations.

Hugh Kennedy, The Prophet and the Age of the Caliphates, 2016. (Link)

The most recent edition of Kennedy's narrative history of early Islam. If you want a roughly 60 page introduction to the era of Muhammad and the Rashidun, this is your best bet.

Michael Cook and Patricia Crone, Hagarism, 1977.

The classic sceptical work referenced in my answer. Though influential, you'd best consult more recent works to get the most currently viable sceptical approaches.