A search of civil births shows a Michael Lyng born on the date you mentioned to father James Lyng and Anne Kehoe ... are you certain about the parent names for your Michael ?
There's a baptism for this Michael in Inistioge Catholic Parish the day after his birth.
A qucik search of the Inistioge baptisms shows a number of possible siblings to this Michael including an Anastasia baptised in 1875, but no Richard that I could see...
In case this is the correct couple - checked for their marriage, luckily took place during civil registration.
Marriage of James Lyng & Anne Kehoe on the 12 January 1866, at the Rower Catholic Church. First marriage for both the bride and groom, father of the groom is John Lyng a farmer, and father of the bride is Michael Kehoe also a farmer. The bride's residence looks like Mungan, the same as that given for place of birth for Michael. I cant quite make out the groom's residence at the moment... looks like 'Can...ol'
The parish record of the parish is under Inistioge parish just gives date, name of the priest Rev. N. Phelan, names of the bride & groom, names of the witnesses James Lyng & Catherine Kavanagh and one residence - Mungan/Mangan