This is just a guess, but the place you're looking for may be Moyaliff (in Irish, Maigh Ailbhe), which looks to be 8-9 miles north of Cashel and 4-5 miles west of Thurles. That's the name of a civil parish (an administrative district), and is also the name of one particular townland within that parish. You can find more info about the civil parish here, where you'll also see links to info about the townlands within the parish: https://www.townlands.ie/tipperary/moyaliff2/
It's a bit hard to tell from the maps at that site and at this next link, but it looks as though the Catholic parish for the area is Ballycahill and Holy-Cross, for which the parish register records are available online back to 1835, and can be found at this link: https://registers.nli.ie/parishes/0243
If the dates you have are right, and if that's the right parish, then you might be able to find a marriage record for Patrick and his wife, although weddings usually took place in the bride's parish, so if she was from Pallasgreen, then the marriage was probably performed there.
You might also try the parish records for Thurles, which go back to 1796. In some cases, if there was not yet a church building in a rural parish, children were baptized in a nearby parish which had one (this happened in my own family). There's an interactive map at the site just mentioned above which will allow you to move on to other parishes. You might come across a baptismal record for Parick there. The O'Dwyer surname (in Irish, Ó Duibhir) originated in mid-Tipperary, so it would be fairly commmon in the area, but you have his parents' names as well..
At the record for each townland at the frst site which I mentioned, there are links to records from the 1901 and 1911 censuses, as well as Griffith's Valuation (mid-1800's), if they are avalable. I checked in the townland of Moyaliff, and there are no records shown from Griffith's Valuation, but in the 1901 census there are two Dwyer families shown as living in the townland, one headed by a Michael Dwyer, born around 1841, and one headed by a Denis Dwyer, born around 1846. They might have been children of another of Patrick's siblings. Both of those families are shown again in the 1911 census, and they had a lot of children, so you might find some Dwyer's still living there today, although on Google Earth it looks as though there is only one homestead still there.
Mockler is also a Tipperary name. It was originally a Norman name, Mauclerc, for which records go back to at least 1210. As the Normans gradually became "more Irish than the Irish themselves", It was Gaelicized as Móicléir, and in modern times it was anglicized as Mockler.
