Reward  in Islam for One Who Loses a Child
Question

What is the reward parents receive for being patient and thanking Allah, when they have an infant or child die?
Praise be to Allah.Virtue of patience in Islam
There are many texts in the Quran and Sunnah which point to the virtue and great reward of those who are patient and state that Allah will give them reward without measure. This includes everyone who bears with patience any calamity with which he is tested. 
Reward for losing a child in Islam
Undoubtedly the loss of a child is one of the greatest calamities for the one whom it befalls. Whoever bears that with patience and accepts the will and decree of Allah will have a great reward by the grace and bounty of Allah. There follow some of these texts, which we hope will be of consolation to you in your time of calamity : 
Allah says (interpretation of the meanings): 

{And We will surely test you with something of fear and hunger and a loss of wealth and lives and fruits, but give good tidings to the patient * Who, when disaster strikes them, say, "Indeed we belong to Allāh, and indeed to Him we will return. * Those are the ones upon whom are blessings from their Lord and mercy. And it is those who are the [rightly] guided.} [Al-Baqarah 2:155-157]
{ … And Allah loves the steadfast.} [Al `Imran 3:146] 
{ Indeed, the patient will be given their reward without account [i.e., limit].} [Az-Zumar 39:10]

And there are many similar verses. 
There are also many Hadiths, including the following: 
Muslim (5318) narrated that Suhayb said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “How wonderful is the affair of the believer, for his affair is all good, and this applies to no one except the believer. If something good happens to him, he gives thanks and that is good for him, and if something bad happens to him, he bears it with patience, and that is good for him.” This has to do with the virtues of patience in general. 
Hadiths on losing a child
There are also Hadiths which speak specifically of losing a child . For example: 

At-Tirmidhi (942) narrated that Abu Sinan said: I buried my son Sinan and Abu Talhah Al-Khulani was sitting at the graveside. When I wanted to go out he took my hand and said, “Shall I not give you some glad tidings, O Abu Sinan?” I said, “Yes.” He said, “Ad-Dahhak ibn `Abd Ar-Rahman ibn `Arzab narrated to me from Abu Musa Al-Ash`ari that the Messenger of Allah (peace and blessings of Allah be upon him) said: “When a person’s child dies, Allah says to His angels, ‘You have taken the child of My slave.’ They say, ‘Yes.’ He says, ‘You have taken the apple of his eye.’ They say, ‘Yes.’ He says, ‘What did My slave say?’ They say, ‘He praised you and said “Inna lillahi wa inna ilayhi raji`un (Verily to Allah we belong and unto Him is our return).’ Allah says, ‘Build for My slave a house in Paradise and call it the house of praise.’” (Classed as sound by Al-Albani in As-Silsilah As-Sahihah, 1408) 
There is a special reward for the person whose child dies but he bears that with patience and hopes for reward. It was narrated from Abu Sa`id (may Allah be pleased with him) that the women said to the Prophet (peace and blessings of Allah be upon him): “Give us a day (to teach us),” So he preached to them and said, “Any woman who loses three of her children, they will be a shield for her against the Fire.” A woman said, “And two?” He said, “And two.” (Narrated by Al-Bukhari, 99 and Muslim, 4786) 
According to a report narrated by Al-Bukhari (1292), Anas ibn Malik (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “There is no Muslim who loses three of his children before they reach the age of puberty, but Allah will admit him to Paradise by virtue of His Mercy towards them.” 
These Hadiths explain that if two or more of a person’s children die and he bears that with patience, then he is promised Paradise and salvation from the Fire. 
Our Prophet (peace and blessings of Allah be upon him) taught us a Du`a to say at times of calamity, in which there is virtue and great reward. Muslim (1525) narrated that Umm Salamah (may Allah be pleased with her) said: I heard the Messenger of Allah (peace and blessings of Allah be upon him) say, “There is no Muslim who is afflicted with a calamity and says that which Allah has enjoined, Inna lillahi wa inna ilayhi raji’oon. Allahumma ajirni fi musibati wakhluf li khayran minha (Verily to Allah we belong and unto Him is our return. O Allah, reward me for my calamity and compensate me with something better than it), but Allah will compensate him with something better than it.” She said, When Abu Salamah died, I said, who among the Muslims is better than Abu Salamah, the first household to migrate to the Messenger of Allah (peace and blessings of Allah be upon him)? Then I said these words, and Allah compensated me with the Messenger of Allah (peace and blessings of Allah be upon him). 

We ask Allah to bless you with patience in the face of your loss, and to compensate you with something better. 
And Allah knows best.