Former Romania defender Chivu will take charge of Inter, where he won three Serie A titles and the 2010 Champions League as a player, after guiding Parma to Serie A safety in his first senior coaching job.
"I thank the club, staff, players and fans for having believed in me and our project," said Chivu on Instagram.
"Together we overcame obstacles and wrote a page (in Parma's history) which will remain in my heart."
Parma later confirmed Chivu's departure, thanking him for dragging the club out of the relegation zone after replacing Fabio Pecchia in February.
That was Chivu's first job in football since leaving Inter's youth set-up last summer.
Chivu has reportedly agreed a two-year contract worth 2.5 million euros ($2.85 million) a season with Inter who have gambled on the novice coach ahead of the upcoming Club World Cup, after failing to get Cesc Fabregas from Como.
The 44-year-old replaced Fabio Pecchia at Parma in February, his first job in football of any form since leaving Inter's youth set-up last summer.
Inzaghi left Inter last week in the wake of the Italians' thumping 5-0 defeat at the hands of Paris Saint-Germain in the Champions League final.
Inter had been gunning for the treble but conceded the Serie A title to Napoli on the final day of the season, were knocked out of the Italian Cup by AC Milan and humiliated by PSG in Munich.
Chivu's first match in charge of Inter will be against Monterrey in the Club World Cup next week, in the Rose Bowl in Pasadena, California.
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