Health Minister JP Nadda said that the "guilty will not be spared".
"Ordered inquiry into incident reported in media involving death of child in Delhi due to Dengue and suicide by parents. Guilty won't be spared," the minister said in a tweet.
The couple, identified as Laxmichandra and Babita Rout, committed suicide by jumping from a four-storeyed building in south Delhi's Lado Sarai.
Delhi Health Minister Satyendar Jain has ordered issuance of show-cause notice to the two hospitals, said a senior state government official.
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The deceased, both natives of Odisha, left behind a one- page suicide note in which they have written that "it was nobody's fault" and that "the decision" was theirs.
Police have initiated an inquest proceeding in the case under Section 174 of CrPC, but no case has been registered so far.
The dengue death toll in the national capital now stands at five although the Municipal Corporation is yet to include three deaths in the official list.