Both sides went into the game chasing qualification for European competition and Giuseppe Biava's early goal put the Biancocelesti on the path to a Champions League place.
Klose doubled the hosts' advantage just before the break and Parma would end the day six points shy of contention for the Europa League, despite Ishak Belfodil halving the deficit by following up his own saved penalty.
A sleepy Parma gave their hosts free reign in the early exchanges, with Stefano Mauri and Hernanes both unlucky not to get past Antonio Mirante, but Lazio took a deserved lead in the 25th minute.
It was a scrappy effort but Biava got under the bar to nudge the ball over the line when no-one had deigned to clear Antonio Candreva's corner to safety.
Biava had registered his first goal in just under a year but Lazio doubled their lead in the 34th minute through a more familiar outlet.
Moments after Parma had appealed unsuccessfully for what appeared to be a handball from Klose, the German striker ran onto an Alvaro Gonzalez pass to coolly nutmeg Mirante.
The Crociati showed faint signs of life just before the break, first through Nicola Sansone, who snatched the ball from Luis Cavanda in a dangerous position.
Michael Ciani put his foot through the ball but Gabriel Paletta should have scored from the resulting corner, the Argentinian drawing two impressive saves from Albano Bizzarri instead.
Klose would not see out the half, demanding to be substituted for an injury and seeing Libor Kozak take his place.
Mirante coped well with a swerving drive from Candreva as play resumed, but Parma would soon enjoy a mini-revival.
With Gianluca Musacci and Marco Parolo seeing good efforts diverted to safety, Gialloblu boss Roberto Donadoni introduced Belfodil for the anonymous Amauri.
And in the 66th minute, Sansone went to ground in the Lazio box under the attentions of Mauri, allowing Belfodil to clip the ball home even though Bizzarri had initially saved his spot-kick.
It should perhaps have been 2-2 four minutes later, but Belfodil rifled the ball straight at Bizzarri following a wonderful cross from Massimo Gobbi.
Mirante raced out of his area to head the ball clear with Candreva rushing in behind the Parma backline, before Kozak inexplicably missed the target from a few feet away, but Lazio were looking good for a third goal.
It would not come, though, and Parma were unlucky not to snatch a point at the death as substitute Dorlan Pabon's ball over the top skipped just beyond Belfodil's stride.