A 78th minute goal from Nicola Sansone cancelled out Andre Pirlo's free kick earlier in the second half, and ensured that their lead at the top of Serie A was reduced to three points, with Lazio leading the chase in second place.
Mirko Vucinic's dummy was intercepted by the hosts and it caused the counterattack which led to the equaliser, and Conte claimed that the trickery was his idea.
"The goal was my fault, because I called the dummy when Vucinic was on the one-on-one," told reporters.
The Juventus coach said that his side deserved to win the match and was pleased that their performance was much-improved compared to their defeat to Sampdoria last weekend.
"Juventus deserved to win considering everything we did today, but were playing against a tough side and give credit to Parma," Conte added.
"I must say that if last Sunday I was angry about the performance against Sampdoria, today I am happy with the opportunities we created.
"On some occasions we got the final pass wrong because we were a little frenetic. We always try to play constructive football and take risks, so it’s a shame we conceded that goal."