Tomasz Maziarz from the Church of Sacred Heart of Jesus & St Cuthbert in Bedford said it was a "profanation as it is a holy area".
Parishioners said the graves, which date back over 200 years, were first attacked on the night of 26 July and then more were vandalised on Monday evening.
Insp Vicky Stoughton from Bedfordshire Police said that "officers will continue to engage with the church to help prevent further offences and identify those involved".
St. Cuthbert's was taken over by the Church of England in the 16th Century and became a Polish parish in 1979.
Gina Marszalek, who is a member of the parish committee, described the attacks on "the sacred tombs" as "very disturbing".
She said the attack on Monday may have involved a hammer or something similar and the perpetrators may have been looking for something or hiding something.
She posted pictures of the damaged graves on social media "to raise awareness and to encourage anyone to come forward".
"I was absolutely amazed and overwhelmed at the response of the local residents who with one voice condemned these very violent and disturbing acts," she said.
She added that "ultimately these are sacred places where somebody's remains have been laid to rest and they should stay that way.
"Unfortunately somebody or a group of people have decided that it is not somewhere they consider to be sacred."
Wojciech Lapinski, 37, who is the Health and Safety coordinator for the parish said he was first informed about the vandalism by the alcoholic anonymous (AA) group which holds meetings at the church.
He said he would like to raise money for extra security at the church including CCTV.
Insp Stoughton said: "Following the initial report, a member of our Crime Bureau contacted the individual who made the report to provide an update on the status of the investigation.
"A further report was received shortly after 12.30pm on 26 August and officers attended the scene within 24 hours to take a statement.
"St Cuthbert's Church and the surrounding area are regularly patrolled by our community policing team."