Pseudo-Sebeos (he is 'pseudo' because this History almost certainly wasn't written by an Armenian bishop named Sebeos) actually didn't mention Heraclius offering to become Khusro II's son. This was instead preserved in the Paschal Chronicle, a Constantinopolitan chronicle from c.629. I looked at Pourshariati's citation here and it turns out she was citing James Howard-Johnston's commentary on the relevant passage from pseudo-Sebeos. However, Howard-Johnston was in fact discussing a letter quoted in the Paschal Chronicle. This is only really a minor mistake, since both pseudo-Sebeos and the Paschal Chronicle mentioned a Roman peace offer in 615 and both contained humiliating terms for Heraclius.

For pseudo-Sebeos, the gist of the offer was basically this: the Roman Empire would remain, but the emperor also stated that Khusro can pick a candidate for the imperial throne, effectively making the empire a Persian client-state. There was also an attempt to appeal to Khusro's better nature by invoking the memory of emperor Maurice, whose murder in 602 precipitated the war, as Khusro had been placed on his throne by Maurice and he then sought to 'avenge' him by making war on his successor. With Maurice having already been avenged by the Heraclian revolution of 608-10 and the murder of the usurper Phocas, the logic goes that Khusro should now be willing to make peace, especially if the Romans were willing to concede so much. The sense of desperation is clearly brought out in the words pseudo-Sebeos put into Heraclius' mouth at this point. The tone is actually a bit uppity, since Heraclius was speaking to a Persian general at this point, but the words themselves tell us a lot about how weak the Roman negotiating position actually was:

For he [God] did not give you victory because of your piety, but because of our impiety. Our sins have brought this about, and not your valour. What does your king seek from me, who does not make peace with me? Does he wish to abolish my empire? Let him not try, because God established it and no one is able to destroy it, save that if it so seems good to God, let the will of God be done.

If he should say: I shall install a king for you, let him install whom he wishes, and we shall accept him. [...] But he [Maurice] had mercy on him [Khusro]. I request the same from him: reconciliation and friendship. I also request from you three things; so heed me. Remove from my land fire and sword and captivity; and from these three things you will find profit, because you will not suffer from famine, nor will the royal taxes be lost. Behold, I shall send to your king a letter with gifts, to seek from him conciliation and peace with me.

The speech most likely was not a genuine speech from 615, but something pseudo-Sebeos crafted in the late 650s, but it was nonetheless one that that captured the spirit of the diplomatic dealings of 615. It did not explicitly mention a master-vassal relationship, but the terms themselves and the evocation of the memory of Maurice says it all - the Romans were losing, badly, and peace whatever the cost was preferable.

This is confirmed by looking at the Paschal Chronicle, which preserved a letter sent by the Senate to Khusro. That in itself says a lot, since it did not come directly from the emperor. Heraclius was after all not seen as a legitimate emperor by Khusro and the events described by pseudo-Sebeos clearly demonstrated how his imperial status was a possible sticking point in the negotiations. Though this letter from the Senate obviously defended Heraclius' position, it was no doubt seen as better for the Romans if this important letter came not from the emperor, but from an institution that was kind of independent-ish from him. The writer(s) of course took care to praise the Heraclian dynasty by mentioning that it was they who had brought down Phocas, Maurice's executioner, so again, like the information preserved in pseudo-Sebeos, the Romans were aware that trumpeting Heraclius' imperial status was a bad thing to do at this stage - diplomatic tact/necessity meant that they had to constantly appeal to Maurice's memory, rather than the standard spiel about how awesome the reigning emperor was.

Anyway, in this letter we have the dramatic statement that Heraclius was willing to be Khusro's son:

We beg your Clemency to hold our most pious Emperor Heraclius as (your) genuine child, for he eagerly wishes to do the service of your Serenity in all things. In doing this, you gain a double glory for yourself, both for your valour in wars and for the gift of peace. And henceforth we shall enjoy peace through your eternally remembered gifts, taking the opportunity to offer prayers to God on behalf of your long good health, and holding your good deed in remembrance through the ages of the Roman state.

Again the terms did not explicitly state that Rome would become a client state, but the signs are there. First of all, becoming a son of the emperor/shahanshah was always seen as a sign of submission: Khusro, only a few decades earlier, was called the son of Maurice, and their resulting co-operation after Khusro's victory in Persia was heavily favourable to the Romans. The more conventional way to describe the relationship would be to see the two rulers as brothers, but here it was made clear that Khusro had the dominant position and that Heraclius/whoever replaced him was to serve the shahanshah. In any case, the results of this embassy are enough of an indicator of the situation: the ambassadors were promptly imprisoned by Khusro according to the ninth-century Short History of Nikephoros. This was as far as I know unprecedented, since ambassadors were normally treated with respect and with a certain degree of ritual even during war-time, at least according to the works of Peter the Patrician and Menander the Guardsman from the sixth century.

So why were the Romans so desperate? 615 was in reality a pretty awful year for them. In 614 the Persians took Jerusalem, an event "worthy of unceasing lamentations" for the Romans and noted by many contemporaries as a disaster of the worst kind. The True Cross itself was seized by the Persians, monastic refugees spread tales of woe to Egypt and North Africa, whilst in Constantinople itself liturgical innovations were introduced by the patriarch Sergius to remind the citizens that God was still with them. Things got so bad that a Heraclian appointee to the patriarchate of Alexandria, John the Almsgiver, attempted to meet the emperor to urge him to make peace, at least according to the patriarch's hagiography written decades later; even if it did not reflect the reality of 614, it seems to me to be a reasonable approximation of public opinion of the time, since presumably the hagiographers would know about the negotiations the next year and how much ground the Romans were willing to give.

In 615 the situation got worse, as a Persian army actually marched all the way to Chalcedon on the Asian side of the Bosphorus. It was at this point that Heraclius began to negotiate with the Persians (see pseudo-Sebeos), which in turn presumably led to the senatorial delegation being sent to Khusro (as recorded in the Paschal Chronicle). It is also worth noting that Heraclius' position was not particularly strong anyway, as he had seized power as an usurper and his rise was marked by civil war in Egypt and massive riots in the Levant. After only 5 years on the throne and a series of major military defeats, his position was by no means secure and peace was certainly desired. As the Persians were winning on all fronts, only such humiliating terms can possibly be accepted, but even then, as we know, the terms were not good enough, as the Persians saw a chance to seize all of the Roman Near East and indeed would be successful in this endeavour.

Most of this information can be found in the narratives of Peter Sarris' Empires of Faith: The Fall of Rome to the Rise of Islam, 500-700 (2011) and Walter Kaegi's Heraclius: Emperor of Byzantium (2003). For analysis, James Howard-Johnston's Witnesses to a World Crisis: Historians and Histories of the Middle East in the Seventh Century (2010) is your best bet. Howard-Johnston is currently working on a new book on this 'last great war of antiquity', so be on the look out for this in the future - he gave a pretty great seminar talk on this recently and I'm certainly looking forward to it based on what I've heard!