There are a few things to keep in mind here, generally speaking. First is - there is a difference between a field army, and the total number of troops.

A total number would include garrisons, militias, and so on - who might not participate in offensive expeditions, and not require full pay and equipment. In contrast, field armies are expensive, and for the Romans tended to be the result of years of training.

That can be seen easily even before 420 AD - the defeat in the battle of Adrianople in the late 300s crippled the eastern legions for years - the numbers involved there were ~15-25,000 roman troops.

Now, let's look at the specifics in 640, and the fall of Egypt. Very important background information here is the final Roman-Sessanid war of 603-628 - see this answer by /u/slin28 for a fascinating description.

What shines through here is that it's a war that lasted over 20 years, in which most of the Roman east had gotten occupied for over a decade, and that ended about a decade before the arab invasions occurred - and then, only by desperately combining all the Roman field armies, abandoning the Balkans, and marching into the Persian heartland. Then, we run into a lack of good, primary sources - but what appears to have happened is that the only sizable Roman force was defeated and broken at the battle of Yarmouk, and Heraclius pulled his forces back - at which point, the forces in Egypt were forced to fight on their own. There was an attempt to take back the province after it was taken - but that expedition failed.

From Adrian Goldsworthy's How Rome Fell: Death of a Superpower, it doesn't sound like all or even much of Rome's army circa 420 CE (or, probably, at most points in its history) could have been mobilized for a single campaign. Military units would be stationed throughout the Empire to keep order and for border wars (for example, wars with Parthia or in Germania). As the Empire continues to destabilize, forces are increasingly used for infighting among Roman generals and politicians. In other words, the entire army was not available for concentrated offense.

Goldsworthy also makes some interesting points about trying to account for the size of the army, even using official records. Two points that stood out to me were: units that exist on paper but not in fact (including officers or regional governors who would try to requisition pay or supply for non-existent troops) and the probability that most military units in the later Empire would probably under strength.

I'm also curious what you mean "Egypt and Spain were conquered... with less than 15k men." Do you mean that that was the army fielded in those campaigns, or that this was the total size of the army in the Eastern Roman Empire at those times? Because if you mean the former, you should consider that comparing 15k as a single field force to 350k as all the forces in an Empire.

How do you know there were 350k soldiers. Can i get a source?

In response to your question, Visigothic Spain was only small part of the late Western Roman Empire and not necessarily a military might on their own, plus fate was not on their side. In the best of times the kingdom was having problems skirmishing with the Basques, Franks and Byzantines. However at the time of the Umayyad Caliphate invasion it was suffering from deep internal strife and the kingdom was fractured into several pieces with those pieces fighting amongst themselves as well.

The inability of the Visigoths to put aside their differences and unite against a common foe was the reason such a small army/raiding force was able to conquer most of the Iberian Peninsula. Had the attack come just a few years earlier or perhaps even a few years later, a force this size would most certainly have been met with a much larger army. An invasion of Spain would have required a much more greater investment of troops but luck was on the side of the invaders. A "perfect storm" one could say.

Initially Berber raiding forces, numbering at the most only a few thousand, were launching attacks on southern Spain mainly to plunder. However seeing almost no resistance from the Visigoths the raiding force was reinforced with several thousand more Berber and Arab forces. The size of this force is arguable but most agree it was not more than 15,000. Embattled Visigoth King Roderic attempted the only serious resistance with a disloyal, poorly equipped and disorganized army. In battle many of King Roderics troops mutinied and King Roderic was killed. This caused the a further power vacuum on the Iberian Peninsula allowing it to be easily conquered piece by peace.