Traditionally these prayers were offered about 11:30 P.M. before Midnight Mass on the last day of the year.
According to the Layman's Ritual 1944 A.D., there is an indulgence attached to saying these prayers on Jan. 31st; plenary if offered before the Blessed Sacrament.
"In a spirit of repentance for our faults and offences of the (past) year the antiphon "Parce Domine" and the psalm "Miserere" are sung before the Blessed Sacrament: and to thank God for His graces and protection the Hymn "Te Deum" is chanted."
The Parce Domine twice before the Miserere and once after.
May God Bless us in the New Year.