Blewitt left Pittsburgh as the school's career scoring leader with 363 points after making 55 of 79 field goals and 198 of 204 extra points from 2013-16. But he has not played in the NFL.
The Bears announced the move on Wednesday.
The Bears let Parkey go after one rough season that ended with a missed field goal off the left upright and crossbar in a wild-card loss to Philadelphia at Soldier Field. Chicago won the NFC North at 12-4 last year.