Fr. Z sets the record straight and thus the case is closed. You can read his entire post on it HERE.
These are some excerpts:
...The SSPX might have a chapel, but they don’t have a parish. Parishes are officially established by proper authority. The SSPX doesn’t have authority to establish parishes.
...The 1983 Code of Canon Law says that:
Can. 966 §1 For the valid absolution of sins, it is required that, in addition to the power of order, the minister has the faculty to exercise that power in respect of the faithful to whom he gives absolution.
§2 A priest can be given this faculty either by the law itself, or by a concession issued by the competent authority in accordance with can. 969.From this we see that priests must have permission of the Church to absolve sins. The Church, by the way, gets to determine how the sacraments are administered.
My Comments: Of course a penitent who is completely ignorant of the fact that an SSPX priest can't validly absolve sins (and also in the case of a priest who isn't validly ordained, or let's say someone who is in the priest's part of the confessional and is pretending to be a priest) may well have the Church supply what is lacking, that one's sins are forgiven, at least through the concept of "perfect contrition."
The sin isn't on the ignorant one who is confessing, but on the SSPX priest who hears confession who has no excuse for not knowing that the Sacrament of Penance isn't valid when he hears confessions unless it is an extreme emergency of someone who is dying. But then even a laicized "defrocked" priest could anoint and hear the confession of a dying person when no other priest is available.