My mom was a control room operator at Beaver Valley and my uncle is a nuclear chemist who has been all over the world. The problem with nuclear is the chance for a huge disaster and the waste. The waste was my uncles main focus. My mom said Homer Simpson is not too far off.
My dad worked for many years on safety and evacuation of a nuclear power plant, worked with the engineers at such plants for decades, and his experience was vastly different. I don't know what position the people your parents are referring to worked, but as for the engineers working inside the plant and handling the reactor and turbines, they are subject to random drug tests, monitoring by supervisors and co-workers, and have significant training and experience.
Most non-supervisory positions inside the control room, such as electricians, require a 3-4 years of apprenticeship plus 2 years of training. That is 5-6 years to qualify for the job, or basically more time than is needed to get a college degree. Further, the vast majority of companies hiring for such jobs require a college degree in addition to the apprenticeship and on-the-job training, so we are talking at least 7-8 years of experience and education for somebody with an AA degree, 9-10 years for somebody with a B.S. degree.
The depiction of Homer Simpson as being in charge of the reactor control is meant to mock the nuclear industry. It's supposed to be funny. "See, the safety of thousands depends on the work of an imbecile because nuclear energy is so unsafe! Ha-hah!!"
You are of course correct that the more operational plants we have, the greater the chances of a serious accident. Most have basically zero knowledge of how nuclear plants, or indeed any power plants, create electricity. The basic process is that water under significant pressure is superheated to well beyond normal boiling temperature, that water than sheds its heat to different water in a separate set of pipes, making that water boil, the boiling water creates steam, the steam turns a huge turbine, and the rotating turbine is wrapped with copper wire. The outer shell of the turbine chamber is lined with magnets. The rotating copper interacts with the magnets, lining up electrons - i.e., creating electricity.
Nuclear plants use nuclear rods with enriched uranium, Uranium[SUP]235[/SUP], as fuel. The U[SUP]235[/SUP] is decaying and emitting a relatively significant amount of radiation in the process and when the protons and neutrons bash into other U[SUP]235[/SUP] molecules, that speeds up the breakdown. The reaction is a mini-chain reaction but never large enough to reach a self-sustaining fission reaction, i.e., a nuclear explosion. (In point of fact, no nuclear reactor could ever produce a nuclear explosion - the engineering to get the plutonium to develop a self-sustaining reaction with the highly-enriched U[SUP]238[/SUP] is so vastly beyond what goes into making a nuclear reactor core that the comparison is frankly stupid.) The low-enriched U[SUP]235[/SUP] in the reactor core (as compared to U[SUP]238[/SUP], which can be converted through a very particular process to U[SUP]239[/SUP], or plutonium) creates a significant amount of heat when the mini chain reaction takes place. Heat = energy.
The biggest risk is the loss of the pressurized liquid coolant either in the reactor core or in the conversion loop, either of which can increase the temperature inside the core to levels which run the risk of causing a break or fracture in the reactor core that releases the highly radioactive materials inside the core, something commonly called a "meltdown." Chernobly involved a unique design flaw where the reactor core exploded due to the creation of highly-flammable hydrogen gas (think Hindenburg).
Three Mile Island occurred when a filter backed up and the technicians tried to clear the clog with pressurized air. The clog interfered with water being cycled to remove the heat (steam) in the reactor core. (The control rods in the reactor are made of cadmium or boron, since those absorb a massive percentage of the radiation and basically work to absorb the radiation and reduce the heat generated. The control rods are electromagnetically controlled so in the event of a power outage, despite back-ups to back-ups, they drop into place and slow the reaction, keeping the reactor cool.)
TMI staff had closed secondary water pumps for maintenance, so when the coolant slowed down and the secondary pumps kicked in to cool the reactor, no water flowed and the temperature continued to build in the core. A valve remained open in a pressure tank to relieve the pressure stemming from the heat build-up, i.e., continued creation of steam in a closed container. A warning light falsely indicated the valve was closed, however, due to a design flaw. The operators thereby did not know that steam was being released for several hours. Releasing steam sounds good in theory, since it relieves some pressure, but the steam is a coolant. Water, even steam, will conduct heat much, much more efficiently than air.
That led to the sequence of events resulting in a "partial meltdown," since water mixed with bubbles began cycling through the reactor core. (The bubbles were present due to the loss of water/steam due to the valve being open.) Introducing air (bubbles) into the core is a significant problem since it does not transfer the heat nearly as well as the pressurized water, so the core temperature began to climb. Once the core reaches a certain temperature, the nuclear material begins to melt. The water is now super-heated and simply cannot remove any more heat. The air bubbles are completely ineffective at transferring or removing heat at such temperatures. The core thereby basically melts part of the containment vessel, releasing significant amounts of radiation.
Most of that release is contained within the exterior containment vessel, i.e., the housing surrounding the core itself, but some is going to be released. That is basically what happened and as is true with every accident, we learn. The facilities in operation in the United States and France (which gets about 70% of its power via nuclear energy) no longer have the design flaws present at TMI. No plant in the United States or Europe uses the Chernobly style graphite rods.
The heat build-up at TMI resulted in a partial meltdown and the release of radiation. That incident is the worst nuclear accident in our nation's history, spanning more than 70 years of nuclear power. More people have died from accidents during construction of solar panels than all who died due to an accident involving nuclear power.
So yes, **** happens. As for nuclear waste - we don't separate the re-usable U[SUP]235[/SUP] any more for reasons that are political more than environmental. More than 90% of its potential energy still remains in the fuel, even after five years of operation in a reactor. The United States does not currently recycle used nuclear fuel but foreign countries, such as France, do. So the easiest solution now is to simply recycle and re-use the fuel - but of course the "caveman alliance" that wants to ban all enrgy production and have us live in caves with a life expctancy of 22 will not let us generate nuclear power. Therefore, we can't use the recycled fuel so instead it sits in storage.
https://www.energy.gov/ne/articles/5-fast-facts-about-spent-nuclear-fuel
Since 1982, all electricity generated by nuclear plants has had a very small tax to finance the storage and treatment of nuclear waste. Those wastes are substantial, but currently the country has a surplus of about $40 billion to handle nuclear waste.
https://earth.stanford.edu/news/steep-costs-nuclear-waste-us#gs.jtnju2
The process right now is to simply store it in steel-lined concrete pools of water or in steel and concrete containers, known as dry storage casks. Right now, we have the money and the resources to continue storing it in that manner for years - decades - to come. But we should recycle and re-use the spent fuel rods to make more energy.