La Russa said Parisi will work as a starter for Memphis.
The 6-foot-4 Boggs, 24, was an impressive 5-1 with a 3.28 ERA for Memphis.
"I've been fortunate," said Boggs, who flew from Memphis to Houston Friday morning to join the Cardinals. "I've had a good year so far. They have a really talented ballclub down there. I've had a lot of guys who know how to play the game around me. That makes it a little bit easier."
"He's very well regarded and he's doing a good job," La Russa said. "He deserves it and we need some help. We're lucky Boggs' day [to start] was tomorrow, so we can use him [Friday]."
Boggs was mostly a reliever at the University of Georgia, but has been almost exclusively a starter in his three-plus seasons in the pros. St. Louis drafted him in the fifth round, No. 170 overall, in the 2005 First-Year Player Draft.
"I've been trying to throw my changeup more [this year] than in the past," he said. "I can see the dividends of doing that. Pounding the strike zone and being confident was always a key."
Boggs, who will wear No. 68, said he received word of his promotion about 11 p.m. CT on Thursday when he returned home after going out to eat following the Redbirds game. The Memphis trainer called and told him to get back to the ballpark -- he was going to the big leagues.
"I wasn't expecting it," he said. "Everybody who plays professional baseball wants that to happen. I was fortunate and blessed enough that it happened to me. I figured I'd been pitching well. I was lucky enough to get the call now."
Boggs made his last start Monday at Albuquerque, throwing only six innings, so he should be available immediately for the Cardinals.
"They told me I'd probably be out of the 'pen right now," Boggs said of his role with the Cardinals. "When they call for me, I'll be ready."
He arrived at his hotel in Houston around noon, giving him time to rest and contact friends and relatives.
"I've been on the phone a lot," he said. "I wouldn't be here if it wasn't for a lot of people back home [in Dalton, Ga.] They been supporting me all my life. So it's been nice."
The Cardinals also announced on Friday that they have moved left-hander Tyler Johnson from the 15-day disabled list to the 60-day DL. Johnson had surgery on his left shoulder.
Right-hander Kelvin Jimenez, who had been optioned to Memphis last weekend, had the option rescinded. He was placed on the 15-day disabled list retroactive to May 31.