I feel for you, as I know that this isn't your fault, nor are any of these things your decisions, since you live with your parents and it's their house (I'm assuming you're under 18?)...But I have to say that you've not got a great set-up or situation for a bird to be living in, for a few different reasons, but the one that is the really urgent is the continued cooking inside of the house with Teflon pots, pans, bakeware, etc.
***It's extremely important for anyone and everyone who owns any type of bird to understand that cooking with ANY pots, pans, bakeware, etc. that have a black colored non-stick coating (most-all with black colored coatings contain PTOA's and PTFE's) or any coating that is made of Teflon should not be allowed ANYWHERE inside of a house/apartment/condo etc. It makes absolutely no difference that your bird is kept in the basement and the cooking with the Teflon cookware is happening upstairs in the kitchen, as the immediately toxic/lethal fumes are well-known to travel to all locations throughout even a very large house and kill birds instantly.
There are many documented cases of bird dying instantly from breathing-in Teflon fumes who were kept in a room on the entire opposite end of the house, on a different floor of the house, and with the door to the room their were in shut and latched. Teflon fumes, much like any and all "fumes", are able to travel anywhere inside of your house, and as I'm sure you know, once your bird breathes-in even one breath of the toxic Teflon fumes he will most-likely die and die instantly, and there isn't anything you can do about it once it happens. So if I were you, I would try to talk to your mother about no-longer using any pot, pan, or bakeware that has any type of Teflon non-stick coating, selling them, and buying a new set of ceramic non-stick pots and pans. They work just as well as the Teflon non-stick pots and pans do, they just don't have the block coating, usually they have a white coating on the inside of them. And she can not only sell her current pots and pans on Craigslist, eBay, Let Go, etc., but she can buy a really large, expensive set of luxury ceramic non-stick pots and pans at either Ross or TJMaxx for a fraction of what even the cheaper Teflon non-stick pots and pans cost...I got a brand-new, 18-piece, ceramic non-stick set of pots and pans at Ross that sold at Bed, Bath, and Beyond brand-new for $299 for only $69! So your mom can actually get a much nicer set of pots and pans probably with the money she gets from selling her old Teflon ones...
And if your mom doubts that the bird will be effected by the Teflon fumes while he's in the basement, all you need to do is do a Google-search for "Teflon kills family's parrots" and many links to the most horrific story of what can happen will pop right up. It's the story of a husband and wife who had owned and bred parrots for their entire lives, and had been very reputable and respected breeders and hand-raisers of Macaws. They kept the baby Macaws in Brooder's in their kitchen, but they also had I believe around 40 other parrots in their large home, some pets/family members, and the rest were their breeder-birds. They had a finished basement where their most beloved pet birds lived, one of which was a 60 year-old Cockatoo, and the other a 30 year old Macaw...One Sunday their baby Macaws in their Brooders just started dropping over dead right in front of them, as did a pet Cockatiel that was flying across the kitchen and literally dropped out of the air onto the floor, dead. They assumed it was due to fumes from something (they weren't cooking at the time, nor was the oven/stove turned on), but they had no idea what was causing the fumes, so they opened every window in the house, opened the doors, turned on all the fans, and started rushing the birds that were in the kitchen and nearest to the kitchen outside, since that's where the birds that first started dying were located. Unfortunately, as the wife was rushing any live birds outside as quickly as she could, her husband was going back to the other rooms of the house where their birds were kept behind closed-door, different bedrooms, an office, a den, etc., and finding dead bird after dead bird all over the house, regardless of how far away they were from the kitchen, or whether they were on a different floor of the house, or whether the door to the room they were in was shut or not...
In the end this couple lose like 45 birds or more, including the 60 year old Cockatoo and the 30 year old Macaw, both of which were downstairs in the basement behind a closed/latched door. The description of the husband carrying the dead 60 year old Cockatoo up from the basement and having to show his wife that he was dead was absolutely heart-wrenching...They soon discovered that the source of the instantly-deadly fumes was due to a new, tiny, ceramic space-heater that they had purchased new a while before this happened, but hadn't yet used because it hadn't been cold enough. They were using it for the very first time that morning, and had plugged it in and set in on the floor of the dining room, which was connected to the kitchen...Turns out that this space-heater contained a Teflon coating inside of it, as many space-heaters do, and that was made of exactly the same type of coating that is used on non-stick Teflon pots and pans...And when this coating hits a high-enough temperature the fumes are released, and obviously seep to every corner of the inside of the home. So a tiny little space heater running in the kitchen on the top floor of this couple's house killed 45 birds that were scattered all throughout their home, on all 3 floors, behind closed doors, including the basement...
The answer to your question about putting your cockatiel in the kitchen while you're gone is yes, it will be fine as far as the Teflon fumes go. These aren't the type of fumes that linger in the air or stick to objects, the Teflon coating must be actively under heat to release the toxic, lethal fumes, and once it's not being actively heated to that temperature anymore, the Teflon stops releasing the fumes. So if no one is going to be in your home at all during this period of time, then it will be fine to keep your Cockatiel in the kitchen...And good for him to get out of the basement...
I hope this basement you are made to keep him in is a nice, finished-basement? As opposed to a dark, damp, cold, dirt-floor basement? There are many potential risks to keeping a bird in a basement that isn't finished, like dampness, mold, fungi, bacteria, insects and other parasites, etc. Not to mention that keeping them in a cage in a dark basement all day long, every day is not at all psychologically healthy or at all stimulating for them either. They need as much interaction and enrichment in their daily lives as possible, as they do have the intelligence of a 3-4 year old human toddler. So please do your best to give him as much direct attention and out-of-cage-time as you possibly can every single day, try to get him upstairs as much as possible, as they do need to get some natural light every day as well. If you have a small cage that you can easily carry him around in, then that's the best way of bringing him upstairs for some sunlight, as well as so he can just see people, hear people talking, etc.