The gemara in Sotah 10b says:
"And Rabbi Levi says: This matter is a tradition that we received from our ancestors: Amoz, father of Isaiah, and Amaziah, king of Judea, were brothers."
In other words, both Yesha'ayahu and his father (and his mother by marriage at least) were also members of the royal family.
This tradition tallies well with what it says in the beginning of Sefer Yesha'yahu (1:1):
"The prophecies of Isaiah son of Amoz, who prophesied concerning Judah and Jerusalem in the reigns of Uzziah, Jotham, Ahaz, and Hezekiah, kings of Judah.
During the time of Amatzyah there was much turmoil. He started out as a mostly good king (Divrei Hayamim 2:25:2) but his downfall began when he worshipped the idols of the Edomites (DH 2:25:14). After that King Yoash of Israel fought and defeated Judea as punishment from Hashem (DH 2:25:21-24). After that there was a coup against Amatzyah as further punishment (DH 2:25:27).
Furthermore, Amotz (and his wife) would have seen the tragic end of King Yoash (Amotz and Amatzyah's father)'s reign: After the death of Yehoyada Hakohen, Yoash and other people of Judea began worshipping idols (DH 2:24:18) and eventually they stoned to death the prophet Zecharyah, son of Yehoyadah and cousin of the king (DH 2:24:21). As punishment, Hashem sent the Aramean army to defeat Yoash (DH 2:24:23-24) and subsequently a coup took place, leaving Yoash dead (DH 2:24:25-26).
Rashi on the above mentioned gemara in Sotah explains that Amotz was also a prophet1, so even if he wasn't scared of the political turmoil that overtook the royal family, he would have likely feared for his life as a prophet, considering what happened to his father's cousin Zecharyah.
Moreover, as a prophet, Amotz might have foreseen the continuing turmoil in the time of Kings Uzziyahu and Yotam:
King Uzziyahu was a good king (DH 2:26:4-5) but made one mistake: He assumed he would be allowed to burn the ketoret in the Mikdash (DH 2:26:16-19) and was punished with tzara'at (DH 2:26:19-21).
And then in the time of Yotam it says:
"He did what was pleasing to the LORD just as his father Uzziyahu had done, but he did not enter the Temple of the LORD; however, the people still acted corruptly." (DH 2:27:2)
Rabbi Ahron Marcus in Kadmoniyot, pg. 136-142 explains that King Uzziyahu, out of his righteousness and prophetic capabilities (R' Marcus explains that "Zecharyahu Hamevin" was the prophetic name of King Uzziyahu), thought he was destined to be one of the prophets that were also Kohanim Gedolim (like Moshe and Aharon) and wanted to bring this about through a hora'at sha'ah (temporary law) which prophets are allowed to enact. For various reasons, Uzziyahu thought that because his grandfather Yoash had lived among kohanim as a child while in hiding from Ataliyah, this tied him into the kehunah and the reason he wasn't a Kohen Gadol was because Yehoyadah wed him two wives (and a KG cannot be married to two women at the same time) and because he cut part of his ties to the kehunah by killing Zecharyah. Yet for all of his various calculations, King Uzziyahu was wrong, and for this he was punished.
But because he was loved by Judea, his punishment of tzara'at - seemingly at the hands of the kohanim - angered the people, and this is what brought upon "however, the people still acted corruptly" in the time of Yotam (as R' Marcus points out, it doesn't say that the people were corrupt in the time of Uzziyahu. From here we see that it was a reaction to what happened when Uzziyahu came to the Mikdash and got tzara'at).
But even if only taken at basic pshat value (that is, that the people were corrupt in the usual sense of the term, and not out of loyalty to the beloved former king), there was still a lot of turmoil going on in those days.
Amotz may have even foreseen what went on in the time of King Achaz, who was an evil king2 (DH 2:28:1-4) but also suffered greatly at the hands of Aram and Yisrael (DH 2:28:5-8), Edom and Plishtim (DH 2:28:17-19) and Ashur (DH 2:28:20) or what happened in the days of King Chizkiyahu, who, though being a good king, was nevertheless attacked by Ashur (DH 2:32:1-22).
Perhaps Amotz even saw Yesha'ayahu's demise at the hand of Menashe, Amotz's great-grandson and Yesha'ayhu's grandson.
To summarize, there were certainly a lot of dark clouds and geo-socio-political and religious struggles and turmoil in those days and a prayer for extra protection from Hashem for their son is very fitting, especially considering that he, like them, was a member of the family who was most in the center of all of these events: the royal family, and especially considering that he, too, became a prophet, which, depending on the era, was, at times, a controversial position to have.
1 It says in the gemara that out of Tamar kings and prophets will come forth; and miut rabim shnayim (the smallest plurality is two)- Yesha'ayhu is one and Amotz is two.
2 Rabbi Marcus even makes him out to be the "Apostmos who burned the Torah", explaining that Apostmos is a scribal mistake and was originally apostatus, apostate, and goes on from there.
