Can only the patriarch give patriarchal blessings?
You have asked a great question. Within the priesthood there are different offices, which are given different responsibilities. Within these offices there are keys granted to some so that they may direct, guide, and preside in their given offices.
A Patriarch is an office within the Melchizedek priesthood. When a Patriarch is sustained and set apart, by a leader who holds the keys, they are now given the authority to act in this office.
Thus, to answer your question, a Patriarch is the only person who is authorized by the Lord to give Patriarchal Blessings.
However, in light of your question, Joseph Fielding Smith once said, as quoted from the Doctrines of Salvation:
“Fathers Can Give Patriarchal Blessings. A faithful father who holds the Melchizedek Priesthood may bless his own children, and that would be a patriarchal (father’s) blessing. Such a blessing could be recorded in the family records, but it would not be perserved in the archives of the Church. Every father who is true to this priesthood is a patriarch over his own house. In addition, children may receive a blessing by an ordained patriarch. A father blessing his own child could, if he received the inspiration to do so, declare the lineage of the child.”
In light of President Smith’s words, it would appear that a father may give a blessing to his children which is similar to a Patriarchal Blessing, and if inspired could declare lineage. However, this would not be a Patriarchal Blessing as given by an ordained Patriarch within the Church, and which is preserved on the records of the Church.