"Millions of Americans have lost their health insurance and are rapidly approaching a point where they must prepare for the possibility of having no health insurance on Jan.1, 2014," he wrote in a letter to Park. "They deserve your sworn testimony before their elected representatives about what went wrong."
While Issa continues to insist that Park stop what he's doing to prepare testimony and answer questions, two Democratic committee members wrote an open letter to Issa on Monday morning, asking him to withdraw the subpoena.
(Read more: Obamacare single-payer a ploy, says ex-GOP Senator)
Ranking member Rep. Elijah Cummings and Rep. Gerald Connolly call the subpoena "unnecessary and misguided" and say Issa's accusations that Park lied about the status of healthcare.gov are "unfounded."
"Mr. Park has been a key leader in around-the-clock efforts to improve the functionality of the HealthCare.gov website, and diverting Mr. Park's energies at this moment could seriously impair those efforts," they wrote.
There's a small but growing segment of the public who agrees, and who want to "Let Todd Work."
Started by a group of technology experts, the "Let Todd Work" website proclaims: "No matter what side of the aisle you sit on, Todd is one of the good guys. Let him do his job."
Park had a minimal role in developing the HealthCare.gov website, which "Let Todd Work" calls "a mess he did not make."
While the site is far from going viral, it and the #lettoddwork hashtag are gaining some traction. As of Monday afternoon, more than 600 people had signed the petition, which appeared online Sunday.
Clay Johnson, one of the engineers behind the protest site, told CNBC in an email that the botched HealthCare.gov site is the result of problems with the hiring, procurement and information technology policies inside the federal government, which "desperately need time and attention from both the executive and legislative branches of government."
"I started 'Let Todd Work' because I've met both Darrell Issa, and Todd Park," said Johnson, who founded Big Window Labs. "And while I've admittedly worked more closely with Park, I think both of these men share the same concern: They both want a government that can design and implement technology effectively and cheaply."
Mike Aleo, who also worked on the "Let Todd Work" site and is CEO of design agency NAV, said, "There's a need ... for answers and accountability, but you don't call the fire chief to the mayor's office to chat during the fire."
—By CNBC's Jodi Gralnick. Follow her on Twitter @jodigralnick.