Reading this is heartbreaking. Teflon items need a big red warning on them for people that don't know.... but I guess some bean counter thought it would hurt sales.
So very very sorry for your loss. I cant imagine how bad you feel.
I am so sorry. The liners often have teflon on them, and when heated they give off fumes which are extremely toxic to birds. Any kind of non-stick pans, bakeware, oven liners, etc. is dangerous. Thank you for sharing this sad information - maybe it will save a parrot's life some day.
I am very sorry to hear of the loss of your babies.
With the loss of your sweet feathered loved ones, this is likely not the time to split hairs regarding their loss. That said, an orange flame, will produce a bit more co2 than a blue flame, which is the design target of the manufacturer (a blue flame). A fully Yellow flame produces the greatest amount of co2.
The co2 death of the three Parrots should have also, seen several of the Human's showing the effects of high levels of co2, including: sleepy, headaches, watery eyes, shortness of breath, etc...
Sadly, placement of most co2 detectors make them useless to protecting one's Parrots as they are commonly located near the furnace. The average home should have a minimum of two co2 protectors.
I am assuming that your liner was near a tight fit into the oven chamber. That configuration would have limited heat above the liner and greatly increased the heat under it. That would easily created enough heat below a coated liner to cause off-gassing.
The only sign of this gas would have been possible shortness of breath of those members who are smokers and /or COPD suffers. Continued or great level of this gas would have had them passing also!
It would be advisable to have all members of your family check for high levels of either in their blood systems. At the bare minimum, call your nearby MedCare facility and ask that question.