Firstly: 

What you are keen to do is one of the noblest of attitudes. 

Allaah says (interpretation of the meaning): 

“So fear Allaah and adjust all matters of difference among you, and obey Allaah and His Messenger (Muhammad), if you are believers”

[al-Anfaal 8:1]

“There is no good in most of their secret talks save (in) him who orders Sadaqah (charity in Allaah’s Cause), or Ma‘roof (Islamic Monotheism and all the good and righteous deeds which Allaah has ordained), or conciliation between mankind; and he who does this, seeking the good Pleasure of Allaah, We shall give him a great reward”

[al-Nisa’ 4:114]

The Prophet (peace and blessings of Allaah be upon him) told us that reconciling between people is better than observing voluntary fasts, offering voluntary prayers and giving voluntary charity. 

It was narrated that Abu’l-Darda’ (may Allaah be pleased with him) said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Shall I not tell you something that is better than the status of (voluntary) fasting, prayer and charity?” They said: “Yes.” He said: “Reconciling in a case of discord, for the evil of discord is the shaver.” Al-Tirmidhi said: It was narrated that the Prophet (peace and blessings of Allaah be upon him) said: “It is the shaver, and I do not say that it shaves hair, but that it shaves (i.e., destroys) religious commitment.” 

Narrated by Abu Dawood, 4273; al-Tirmidhi, 2433. classed as hasan by al-Albaani in Saheeh al-Tirmidhi. 

Islam has permitted telling lies in order to achieve this great purpose. So it is permissible for you to tell each party that the other has said good things about them and praised them so as to encourage reconciliation. This does not come under the heading of lies that are haraam. 

It was narrated from Umm Kalthoom bint ‘Uqbah ibn Abi Mu’eet that she heard the Messenger of Allaah (peace and blessings of Allaah be upon him) say: “He is not a liar who seeks to reconcile between people and says good things.” Narrated by al-Bukhaari, 2490 

Secondly: 

For Muslims to forsake one another is a haraam action. You can remind both parties of the texts which indicate that. These include the following: 

From the Holy Qur’aan: 

(a)The verse in which Allaah says (interpretation of the meaning):

“And obey Allaah and His Messenger, and do not dispute (with one another) lest you lose courage and your strength departs, and be patient. Surely, Allaah is with those who are As‑Saabiroon (the patient)”

[al-Anfaal 8:46]

(b)The verse in which Allaah says (interpretation of the meaning):

“And hold fast, all of you together, to the Rope of Allaah (i.e. this Qur’aan), and be not divided among yourselves, and remember Allaah’s Favour on you, for you were enemies one to another but He joined your hearts together, so that, by His Grace, you became brethren (in Islamic Faith), and you were on the brink of a pit of Fire, and He saved you from it. Thus Allaah makes His Ayaat (proofs, evidences, verses, lessons, signs, revelations, etc.,) clear to you, that you may be guided”

[Aal ‘Imraan 3:103]

From the Sunnah of the Prophet (peace and blessings of Allaah be upon him): 

(a)It was narrated that Abu Hurayrah (may Allaah be pleased with him) said: The Prophet (peace and blessings of Allaah be upon him) said: “Do not hate one another, do not envy one another, do not turn away from one another. Be, O slaves of Allaah, brothers. It is not permissible for a Muslim to forsake his brother for more than three days.”

(Narrated by al-Bukhaari, 6065; Muslim, 2559. 

(b)It was narrated that Abu Ayyoob al-Ansaari (may Allaah be pleased with him) said: The Prophet (peace and blessings of Allaah be upon him) said: “It is not permissible for a Muslim to forsake his brother for more than three days, each of them turning away when they meet. The better of them is the one who gives the greeting of salaam first.”

Narrated by al-Bukhaari, 2757; Muslim, 2560. 

(c)It was narrated that Abu Hurayrah (may Allaah be pleased with him) said: The Prophet (peace and blessings of Allaah be upon him) said: “The gates of Paradise are opened on Monday and Thursday, and everyone who does not associate anything with Allaah is forgiven, except a man who has had an argument with his brother. It is said: ‘Wait for these two until they reconcile, wait for these two until they reconcile, wait for these two until they reconcile.” Narrated by Muslim, 2565.

Al-Nawawi said: “Wait for these two” means wait until they have reconciled and love has been restored between them. 

(d)It was narrated from Abu Kharaash al-Sulami that he heard the Messenger of Allaah (peace and blessings of Allaah be upon him) say: “Whoever forsakes his brother for a year, it is as if he has shed his blood.” Narrated by Abu Dawood, 4915. Classed as saheeh by al-Albaani in Saheeh al-Targheeb, 2762.

These texts indicate that it is forbidden for a Muslim to forsake his brother by not greeting him with salaam and by turning away from him for more than three days, so long as that forsaking is not for a shar’i reason and is not being done in order to achieve some purpose, in which case it is permissible to forsake him for more than three days. 

See also question no. 21878 

You have to remind these two disputing parties of these verses and ahaadeeth. Try to reconcile their points of view, and encourage them to forget their differences. You could address each of them directly, or you could write these texts on a piece of paper and give it to them to read. 

We ask Allaah to help you to do that which is good. 

And Allaah knows best.