Former Liverpool man Jones replaced Ben Williams in goal and things were looking good when his side led 2-0 thanks to goals from fellow Australian James Meredith and Devante Cole.
However Jones was beaten when left-back Meredith was unfortunate to see a header loop into his own goal, and Blades striker Billy Sharp drew his side level after seeing his initial effort well blocked by Jones.
"I thought Brad did well. He is a positive keeper who wants to come for things and he also showed that with his kicking," Parkinson told the Telegraph & Argus.
"It was harsh on Ben (Williams), who has been tremendous for us.
"Ben is not pushed to one side. They are both competing. We conceded a poor goal last week but the way he responded was great."
Jones will hope to keep his place when Bradford City takes on Peterborough United this weekend at Valley Parade.
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