Where have they been mentioned?
Allah mentioned them once in the qur'an:
Indeed, those who have believed and those who were Jews and the Sabeans and the Christians and the Magians and those who associated with Allah - Allah will judge between them on the Day of Resurrection. Indeed Allah is, over all things, Witness. (22:17)
This verse can't be used as a clear evidence for considering the Magians or Zoroastrians المجوس as ahl-al-Kitab as it quotes them as an example of among those who at least had flaws in their belief as they are taken among the opposite examples to the (right) believers.
While Allah quoted the three first of these non-Muslim groups among the believers elsewhere:
Indeed, those who believed and those who were Jews or Christians or Sabeans [before Prophet Muhammad] - those [among them] who believed in Allah and the Last Day and did righteousness - will have their reward with their Lord, and no fear will there be concerning them, nor will they grieve. (2:62)
Which he didn't explicitly for the Zoroastrians.
Furthermore there's a hadith in al-Muwatta' explaining why the sahabah took Jizya from them similarly as the ahl-al-Kitab:
Yahya related to me from Malik from Jafar ibn Muhammad ibn Ali from his father that Umar ibn al-Khattab mentioned the magians and said, "I do not know what to do about them." Abd ar-Rahman ibn Awf said, "I bear witness that I heard the Messenger of Allah, may Allah bless him and grant him peace, say, 'Follow the same sunna with them that you follow with the people of the Book . ' "
This seems an indication that they are excluded form the ahl-al-Kitab as else why should one explicitly ask or order to treat them similar to those?
These evidences created a certain conflict. On whether or not the Zoroastrians are ahl-al-Kitab!
There's as far as I know a consensus among -at least sunni scholars- that it is not allowed to marry their women nor eat the meat of their slaughtered animals.
What is the definition of people of the book
As for people of the book there's a certain consensus that we are allowed to marry their women (under certain conditions and to eat from their food (under certain conditions) as stated in:
This day [all] good foods have been made lawful, and the food of those who were given the Scripture is lawful for you and your food is lawful for them. And [lawful in marriage are] chaste women from among the believers and chaste women from among those who were given the Scripture before you, when you have given them their due compensation, desiring chastity, not unlawful sexual intercourse or taking [secret] lovers. And whoever denies the faith - his work has become worthless, and he, in the Hereafter, will be among the losers. (5:5)
The third special thing that usually identifies ahl-al-Kitab is the payment of jizyah!
As for the hadith adding them to those who need to pay Jizya some scholars considered it also an evidence to generalize the payment of Jizya on all the non-Muslims so it is not limited to the people of the book under Muslim regency.
So are they among people of the book?
From all this it is clear that it is somewhat doubtful that they are among the people of the book. And therefore the position of the majority of scholars is that they actually are worshipers of the fire and that their belief is that there's darkness and light some even said that the origin of المجوس is النجوس as they used in their worship najassah (see for example in tafssir al-Qurtubi or in at-Tahriri wa a-Tanwir where ibn Ashur explained a bit more about the origin of this religion)
Nevertheless some scholars said that they are among the people of the book and that their book has been lifted based on a statement of 'Ali ibn abi Talib which imam al-Baghawi quoted in his tafsir (see here in Arabic) on the verse about the payment of Jizyah (9:29):
In the following I'll translate from Arabic as these translations are of my own take them with the necessary care!
واختلفوا في أن المجوس : هل هم من أهل الكتاب أم لا؟ فروي عن علي رضي الله عنه قال : كان لهم كتاب يدرسونه فأصبحوا ، وقد أسري على كتابهم ، فرفع من بين أظهرهم .
They (also) differed on their position about Magians: Whether they are people of the book or not? As it was narrated from 'Ali ibn abi Talib may Allah be pleased with him that he said; They had a book which they used to study, and one day they "woke" up and their book was lifted (isra'), so that it was lifted from among them.
واتفقوا على تحريم ذبائح المجوس ومناكحتهم بخلاف أهل الكتابين .
While they agreed about the prohibition of their slaughtered animals (dhabihah) and their marriage (marrying their women) unlike the people of the book.
This statement is rather unique and it quotes a narration without giving a source which comes from a scholar who is known to be among the bigger hadith scholars among the mufassireen. So we may take this narration as a "khabar" which is more likely a comment or an unsourced input.
You may also find this fatwa answering your question in Arabic helpful: islamweb #167861.
