This question has actually been the center of a massive historiographical debate in medieval and early modern studies. (How does it feel to be asking the brilliant questions? :D ) I gave an overview of it in this earlier answer, if you're interested. It's more focused on the historiography than the historical evidence, though, so there's a lot more to be said for sure.

As usual, /u/sunagainstgold 's answer is beyond comprehensive in offering the current state of play with academic responses to this question, so I don't intend to add to that further. However, my PhD research at the moment is focused on early modern epitaphs which appear in manuscript, so I hope I can add a little coda to this in terms of literary evidence for mourning children.

Manuscript epitaphs are a distinct phenomenon to engraved ones, because instead of paying someone to engrave something on a stone (with all the issues of cost and space limitations that engenders), you can compose an epitaph in manuscript with few practical limitations.

Having found and catalogued literally hundreds of late Elizabethan and early Jacobean epitaphs, I have found remarkably few for children. Some are famously heartfelt - Ben Jonson's epitaphs for his children definitely indicate the same sense of profound loss that we would expect from any parent today. However, lots of others I have found have a gently humorous tinge to them. For example:

As carefull mothers to there beds doe lay

theire babs wich would too longe the wantons play

So to spend my youth in swadlinge crimes

nature my nurse layd me to bed betimes.

BL MS Add. 30982 f2r

In this manuscript, as well as others, I have found numerous 'occupational' epitaphs, where the subject's occupation is extended into their experience of death, making death feel like an everyday, lighthearted occurence. See:

Here lyes Tom short the king of fellowes

Whom in his time was a mender of bellowes

But when he came to the howre of his death

Hee that made bellowes could not make breath

BL MS Add. 30982 f36v

This is still something I am working on, but it looks like children are being treated in the same way as the adult community here, where their 'occupation' (baby) and the activities associated with their daily lives influence the experience of death in the epitaph narrative. This places children as participants in life, death, and community on a par with adults. There is also the sense that humour is being used here to 'laugh off' a death.

As Marcy L. North discusses in her essay on anonymity, epitaphs for the Crown Prince Henry get clustered up fairly anonymously with deaths for other children in manuscript. These epitaphs are often recycled to stand in for a new deceased so that one epitaph for a child may be taken to represent many different deaths. This is also true of adult epitaphs, so there is a similar usage of commemorative verses, even if they are less common than epitaphs for adults.

The evidence I have acquired so far doesn't seem to indicate that grief for a little one is lessened (as Stone would tend to suggest) or that children are seen exclusively as 'little adults' - there is definitely evidence to indicate that something precious and innocent has been lost compared to adults. See for example, the epitaph for this five year old boy:

Thy Sweete desire to praise thy God

thy tender loue to parents deere

Thy nature milde to euerie one

remaines aliue, though corps lie heere./

Viuit post funera virtus

obijt Ianuarij i587 aetatis suae 5./

Cambridge Add MS 9221, f104r

As far as commemorative verses would suggest, the loss of a child is still regarded as the loss of a community member, they are mourned in the same way as adults (though the loss of a child still has the extra sting of lost potential and a dead innocent) and parents still grieved deeply at the loss of an infant in spite of a high child mortality rate.

Returning to your original question though, yes, parents were more 'likely to accept' the death of a child in so far as they would have accepted this as a strong possibility with the birth of any child. This doesn't mean that they would 'accept' a death in the sense that it would be shaken off as a trivial event. In one of his letters, John Donne talks about having lost 'half a child' (miscarriage) with more of his children sick (one dying), a despondent wife, and a deep despair of his own at such loss.

It's definitely complex to try and assess emotional responses to humans who lived hundreds of years ago, but the evidence seems to suggest that child mortality was 'accepted' as a commonplace, but grieved all the same.

edit: crappy formatting