There are a number of common faulty arguments enunciated by many of those who reject these sacred images, and I would like to list a few for the forum's consideration and further study.
Firstly, I would like to mention, the iconomachs argue that the appellation "Ancient of Days" applies exclusively to the Son, and does not apply to the Beginningless Father. This is false and contrary to the revealed teachings of the God-bearing fathers. The "Ancient of Days" is an appellation pertaining to the Godhead. In order to bolster their argument that the Son is exclusively named the "Ancient of Days, the iconomachs are fond of citing Church hymns such as ones found on the Feast of the Meeting of the Lord:
The Ancient of Days, who in times past gave Moses the Law on Sinai, appears this day as a babe.
As everyone can readily see, the Ancient of Days existed in the time of Moses, prior to the Incarnation, so obviously this appellation pertains to the Divine Nature of Christ, and not His humanity. If it pertains to the divinity of Christ, then it likewise pertains to the Father and the Holy Spirit as well, for they share the same divine essence (homoousian) as the Son.
Allow me to quote two eminent saints to illustrate my meaning:
St. Gregory of Nyssa, Against Eunomius, Book VII:
[T]hat in this case the Father is God and the Son is God; that "just," and "incorruptible," and all those names which belong to the Divine Nature, are used equally of the Father and of the Son; and thus, if the divergent character of appellations indicates difference of natures, the community of names will surely show the common character of the essence.
St. Symeon the New Theologian, The Discourses:
If there is need to state anything more precisely, that which the One is, the other Two are as well. For the Three are in the same and are thought of as one Essence and Nature and Kingship. If a name is attributed to One, it is by nature applied to the others, with the exception of the terms Father, Son, and Holy Ghost, or the terms beget, begotten, and proceeding, for these alone indisputably apply to the Holy Trinity by nature and in distinctive fashion.
Many saints referred to the Father as the "Ancient of Days," including your patron, the most-lovely Mar Ephrem!
St. Ephraim the Syrian, Selected Works, Vol II. p. 511:
For when [God whom we have called] a mirror was incapable of old age, and the (Jewish) people incapable of the truth, He took to Himself old age for the instructing of the faithless: and since king and old man and child were become effeminate, He put on old age; as a venerable old man did He judge those iniquitous persons who were effeminate in sin. The Being that waxeth not old put on old age to teach by parables concerning His Son and His Beloved. By the mask of old age He shewed His Fatherhood to teach that He hath a Son, the Son of Man, Whom Daniel saw standing before the Ancient of Days, Who did away with mortal kings, and made Himself a King in the Son of the King Immortal. If it had been One only that was sitting, then had there been one seat; but for this reason he saw not one seat, but seats. He shewed that there was an Assessor with Him, and a Son to the Ancient of Days. The thousand thousands whom Daniel saw, them alone did he see standing: to the Son of Man he did not ascribe standing, because He is not a minister.
St. Epiphanius the Five-Tongued of Salamis, Panarion Vol. III
14,3 This Father, Son and Holy Spirit has always vouchsafed to appear in visions to his saints, as each was able to receive [the vision] in accordance with the gift which had been <given> him by the Godhead. This gift was granted to each of those who were deemed worthy, sometimes to see the Father as each was able, <sometimes> to hear his voice as well as he could. (4) When he said by the mouth of Isaiah, “My beloved servant shall understand,” this is the voice of the Father. And when Daniel saw “the Ancient of Days,” this is a vision of the Father. And again, when he says in the prophet, “I have multiplied visions and been portrayed by hands of the prophets,” this is the voice of the Son. And when, in Ezekiel, “The Spirit of God took me” and “brought me out unto the plain,” this refers to the Holy Spirit.
St. John Chrysostom, Commentary on Psalm 110
Let us, however, resume once more our theme. The Lord said to my Lord, Sit at my right hand. Do you see the equality of status? Where there is a throne, you see, there is a symbol of kingship; where there is one throne, the equality of status comes from the same kingship. Hence Paul also said, “He made his angels winds, and his servants flames of fire. But of the Son: Your throne, O God, is forever.” Thus, too, Daniel sees all creation in attendance, both angels and archangels, by contrast with the Son of Man coming on the clouds and advancing to the Ancient of Days. If our speaking in these terms is a problem for some, however, let them hear that he is seated at his right hand, and be free of the problem. I mean, as we do not claim he is greater than the Father for having the most honorable seat at his right hand, so you for your part do not say he is inferior and less honorable, but of equal status and honor. This, in fact, is indicated by the sharing of the seat.
As everyone can readily see, St. John Chrysostom is describing the Son of Man seated on the throne at the right hand of the Father, the Ancient of Days, as was seen in the night visions of Daniel the prophet. Notice that St. John speaks of the Son "sharing the seat". Now cast your eyes to the right at my avatar, and you will see a holy icon of the Most-Holy Trinity, which is over 500 years old from Greece, at the Benaki museum in Athens. Notice how in this icon the Son is sharing the seat with the Father, just as the divine Chrysostom describes.