Farbrace, who moved from Sussex at the end of the season, has not yet officially started in his new role at Lancashire but delivered the news to Anderson at a meeting last week alongside chief executive Richard Kenyon, who has also recently been appointed.
The news was first reported on Wednesday and Lancashire did not comment officially until Sunday. At a meeting with members on Monday, the club said they would look into arranging Anderson a farewell.
"It shows a lack of leadership at that club," said Broad, whose combined total of Test wickets with Anderson stands at a record 1,039 in 138 Tests.
"It shows a lack of doing it the right way.
"Paul Farbrace is new, but where is the chair? Where is the board demanding that this is done in a better way?"
Speaking publicly for the first time since the decision on Sunday, Anderson, who also had his England career cut short in 2024 in similar circumstances, expressed his sadness at the situation but stopped short of retiring.
Other counties are understood to be interested in signing the veteran.
"I get the thought process, 'we could probably get two half-decent cricketers for the one and losing that salary'," Broad added.
"It doesn't surprise me that some counties have already been on the phone and seeing what his interest levels are.
"I don't know what his next steps are. I think his statement was quite interesting.
"There will be a lot of counties that want him. He offers a huge amount, he's still one of the best bowlers in the country."