This is the opening and ending theme of "بچههای مدرسه والت" / "Bachehaye Madrese Walt" ("Walt School Kids"), formerly titled "مدرسه عشق: قصهٔ دل" ("School of Love: the Story of Heart"). This anime is the Iranian version of 愛の学校クオレ物語 / Ai no Gakko Cuore Monogatari, a 1981 Japanese anime based on Edmondo de Amicis's 1886 novel "Cuore, Libro per i Ragazzi".
The soundtrack of the Iranian version is different from the original Japanese soundtrack. Actually, it is borrowed from the fourth episode, where street musicians are shown playing this Walz and Enrico's parents are dancing on it. If I understand correctly (the Google translation of) what is written on the Farsi Wikipedia page, this sequence was censored by the Iranian TV but this accordion music was kept as the opening/ending soundtrack. For those who can read Farsi (I can't), more details are given in this video.
According to IMDB, the original opening and ending songs were composed by Yasushi Akutagawa and arranged by Katsuhisa Hattori. They were great composers that were perfectly able to compose such a waltz for accordion. Do you have any evidence other than the video that this is an older song, and was not composed by them? The video is misleading. If you watch carefully, the accordion player is not playing what we are hearing.
You can find the downloadable music score and MIDI file here: https://musescore.com/user/33055012/scores/6494225
(FWIW credited composer is Katsuhisa Hattori but that page is not a solid reference)
You can also listen to a good quality (alternate) version of the song on soundcloud.