I think Melkites are parallel with the Greek Orthodox Church of Antioch. Look at how many Patriarchs there are of Antioch, though.
There are five patriarchs in Antioch.
Let me see if I understand the question correctly.
If you are asking about the 5 patriarchal sees (you mentioned Apostolic Sees), then (I have highlighted the Catholic ones in bold)
Rome: only Catholic
Pope
Constantinopole: only EO Greek (there was a Latin patriarch, now extinct). There is also an Armenian patriarch but he won’t not be an independent patriarch like EO ones, being dependent on the Catholicos. As far as I am aware there are no Eastern Catholics in Constantinople, only Roman rite, largely expats.
Antioch: (i) EO Greek (title: Greek Orthodox Patriarch of Antioch and All the East) ; (ii)
Melkite (title: the Patriarch of Antioch and All the East, Alexandria and Jerusalem) - Catholic counterpart of (i) who covers the Greek Eastern Catholics in Alexandria and Jerusalem as well; (iii) Jacobite Syriac (title: Patriarch of Antioch and All the East)- Oriental Orthodox; (iv) **Catholic Syriac **(title: Patriarch of Antioch and All the East of the Syrians) - Catholic counterpart of (iii) and covers the Syro-Malankarese Catholics in India as well; (iv)
Maronite (title: Patriarch of Antioch and the Whole Levant) - Catholic with no EO/OO counterpart. (there was a Latin patriarch, now extinct). All five patriarchs are based in Damascus except the Maronite & Syriac Catholic (iv) based in Beirut. All five recognises the same Apostolic Succession until 6th century
Alexandria: (i) Coptic - Oriental Orthodox; (ii)
Coptic Catholic - Catholic counterpart of (i) and covers the Ethiopian rite Catholics as well; (iii) EO Greek. The head of the Greek Catholics in Alexandria is the Melkite patriarch.
Jerusalem: only EO Greek. There are also a
Latin and an Armenian patriarch but they are not independent, being dependent on the Pope and the Catholicos respectively. The head of the Greek Catholics in Jerusalem is the Melkite patriarch. Armenian Catholics are under the Armenian patriarch represented by a local patriarchal vicar.
If you wish to add the following ancient sees
Babylon: (i) Assyrian - neither EO or OO. Neither do they like to be called Nestorian; (ii)
Chaldean - Catholic counterpart of (i) and covers (correct me her Thomas48 if I am mistaken) the Syro-Malabarese Catholics in India as well.
Armenia: This one is a little confusing. (i) Armenian Supreme Patriarch with the title of Catholicos of all Armenians (ii)
Armenian Catholic Patriarch of Cilicia, Catholic counterpart of (i) based in Beirut as most Armenian Catholics are in Lebanon & Syria. There is also a Catholicos of Cilicia dependent on (i) whose title outranks patriarchs in the Armenian Church. (Cilicia was an ancient region in SW Turkey, no longer having any Christians)
Ethiopian & Eritrea: There is is an OO Abuna in each country with the rank of patriarch independent of Alexandria in the same way as the EO patriarchs are of Constantinople (ie., administratively independent, doctrinally dependent but any independent course would be dependent on the politics and personalities involved). Heads of neither
Catholic counterpart has the rank of patriarch, being only archbishops.
India: (Thomas48 can help me out here) There is a Catholicos of the Malankarese, OO dependent on Jacobite patriarch. There is also a Mar Thoma church (more a Malankarese offshoot engineered by the Anglicans). Catholic counterpart for both is the
Syro-Malankarese dependent on the Catholic Syriac patriarch. The largest of the St Thomas Christian Churches would be the Catholic
Syro-Malabarese. Their orthodox counterpart would be the very tiny (<1000 members) Ancient church of the Near East, who are basically Assyrians. Both Catholic Churches are headed by Major Archbishops, one rank below patriarch.
Sinai: St Catherine Monastery is technically an autocephalous EO Church and can be considered the only Orthodox Church without a Catholic counterpart.
Eastern Europe
Here, the only division is within EO, between the Ecumenical Patriarch and the Russian Patriarch vying for control/influence over the Eastern European churches.
The only place with a significant Catholic presence is Ukraine, where the
Ukrainian Greek Catholic Church (meaning Byzantine Rite, not that they use much Greek) is headed by a Major Archbishop. Their long running request for a patriarchate has long been a sore point with the Vatican. The EO in Ukraine are resented by three separate churches, not counting the Orthodox church that broke away from the Ukrainian Catholics.
All over Eastern Europe, there would be a mix of EO Byzantine majority and small Catholic Byzantine minority of the same rite (Catholic Byzantine Rite majority only in Western Ukraine). Even the small Ruthenian Catholic Rite (who are actually Ukrainian) would have a EO counterpart - just not by that name.
The only Byzantine Rite without an EO counterpart is the Italo-Albanian Rite, Byzantines left behind in Sicily after the Byzantine temporary conquest under Justinian. They only just got a bishop after over a millennium without one.
I think that covers it all. Always wanted to list them all down. I find the situation in Antioch, India and Ukraine especially interesting, what with schisms, counter-schisms and sub-schisms all over