Hi Peggy
I have dealt with that parish a bit.
No, is the short answer.
It's records are not online.
You will only get them 'online' on the rootsireland site that you are already using.
The issue that you are coming up against now is that when you go back to that time in Ireland it was RARE for the parent's names to be on the church marriage records.
The odd time, if you are lucky, you will get a father mentioned but not much more than that.
So, it's not that the parish records are incomplete, it's just that's how the records were done at the time.
First, did Michael and Bridget have witness' at their marriage ?
Were these relations ?
Next ~ draw up a list of the children of Michael & Bridget (in chronological order) and post the list here ~ naming traditions are your best way to get the parents names of Michael & Bridget.
Your best bet is, now that you have a month's subscription to rootsireland and can get up to 1000 transcriptions then to go in and see are there other Regans in the same parish.
So, what did Michael work at ?
Did he stay in Galway City ?
Have you found him, Bridget or any of their children on the 1901 / 1911 Irish Census ?
If they are on the census, look for their neighbours, see are any of the neigbours possibly related.
You will need to tray and see did Michael & Bridget have younger siblings whom may have married after 1864 when our civil registration began.
If you got a civil marriage for say a younger brother of Michael then THIS would have their father's name and occupation on it.
If there is anything more that you can tell us, then please do and we can try and help you some more !
Tara