The 26-year-old went off holding his troublesome shoulder, but after hospital treatment the thumb proved a more serious concern.
Wales coach Steve Hansen said: "It's a blow for Sonny after his broken arm but he is going to have to do what Kevin Morgan has done so often and keep bouncing back.
"Hopefully, sooner or later he will have a run with no injuries. Other than Sonny, everyone is okay."
Tom Shanklin may be the natural replacement for Parker - he came on for the injured centre and promptly scored two tries in Dublin.
Other options are moving Gareth Thomas from full-back to his preferred centre role, or recalling one of the Lions duo of Dafydd James and Mark Taylor.