Well, I think you should do some searches in the forums for roaches (Dubia and Turkestan) and compile all of the posts you can to give a truly wide spectrum from many individuals on their opinions of roaches vs crickets. If your parents see the sheer number of people that love their roaches that alone might be enough for them to give them a chance.
But, what you asked for....
Crickets stink, they escape because they can jump quite high compared to their body size, they stink, they make lot's of noise, they stink, they live for a very short time and they stink.
Now to break down those aspects.
Escaping: crickets can jump high and fast. They may end up on your arm, in your face, on your shirt, or outside of the container. When you try to grab em, they jump again and make a beeline to the nearest dark hiding space in your room they can find. They will eat ANYTHING they can chew and that includes clothing and carpets. If you don't believe me, get a piece of scrap carpet and throw into a bin with 1k crickets. It will be falling apart in about a week.
Then they chirp and chirp and chirp all night long driving you insane. As soon as you get up and try to track down where they are, they go silent and wait patiently like a sniper in the jungle. As soon as you lay back down... chirp chirp chirp. The only good news if they escape is that they are very short lived. You will probably only have to deal with that one cricket chirping for about two weeks before it dies off. But then, you probably had more escape at that point and the cycle continues. Roaches (aside from a climbing species like Lobsters) just won't escape unless you do something stupid like knock the whole bin over on its side. They can't jump or fly (at least not the common feeder species) and thus their only means of escape is total user error while can be prevented completely. If one were to escape, then it would disappear into a dark spot like a cricket and stay there silent as can be and it would leave your clothing and carpets alone. Roaches are quite picky about what they eat contrary to popular belief. They also require consistently high temps above 80 and plenty of moisture and food to live and breed. Usually if one escapes, you will find it dead a few days later somewhere because the humidity wasn't high enough for it to survive.
Noise: I touched on this in the above paragraph, but that is only one cricket. With 1000 in a bin at a time, it's a heavy metal concert. Just constant chirps overlapping with seemingly no rhythm. And because the frequency of the chirps they produce is so high, the sound pierces through walls readily thus driving you insane even if they are on the other side of the house will all the doors inbetween closed. You will always hear em. Roaches, you won't hear, except for a slight "rain like" rustle when you first enter the room and they are disturbed. That sound you hear is strictly them scurrying into the eggcrates and ceases pretty quickly.
Stink: This is what crickets were put on this earth to do. Ok, not really, but you get the idea. Maintenance is critical to reduce the stench and must be performed daily, sweep out the poo and removing the dead bodies. If you don't remove the dead bodies, it causes a chain reaction of deaths due to the stink. The actual stench doesn't kill them (it might kill you though), but the gases being released that contributes to the stinking suffocates out the other crickets and then more bodies pile up and you get more stench. One bin of crickets can stink up an entire room easily. Roaches, they have a very slight odor depending on species, but usually it is barely noticeable with a few exceptions for certain species. You only have to clean the roach bins every 3-6 months roughly and they won't stink at all, certainly won't foul up an entire room like crickets will do after one week. I just cleaned out one bin of 3k roaches that hadn't been cleaned in 8 months. It started to smell a tad bit when I first took the lid off. I couldn't smell them in the room, only when I first opened the lid, so I figured I would give them a cleaning. I think some moisture got built up in their and soaked the frass (the poo) a bit because it was a bit like an ammonia type of smell. There weren't very many dead ones at all, which we go to now...
Life Cycle: Crickets live for roughly 2 months total. This can be altered slightly by temperatures, but 2 months is the basic time frame from the day they hatch to the day they keel over and start stinking to the 3rd degree. About 6 weeks from hatching, they start maturing into adults. They already stink at this point a lot! But now they can make noise too! So not only do you deal with them stinking for another two weeks, you also have to listen to them! Not fun. A short life cycle may not seem like a big deal, but it really is. For one, they have a higher metabolism than roaches and will eat up more food in the same amount of time as a roach, which means it takes more to keep em alive. They grow and die so fast, you usually have to buy extra (or breed lots of extras if you breed them) in order to make sure you have enough to use as feeders. Ultimately a lot of them go to waste and die off before you use them, unless you buy small amounts, but then that means a lot of frequent online orders which costs lots of money in shipping, or many trips to the petstore which costs gas. It's a hidden cost of the feeders people don't think about. Most roach species live for anywhere from 12-24months (18 is about the average) They grow slower and eat a smaller amount of food than crickets during the same time period, thus less is used to keep them alive and breed em. Since they aren't always dieing off in large batches, there aren't many dead bodies to decay and cause the stink. When you want efficiency, roaches are the way to go. Since roaches eat less, we can use less plastic bags from dog food, produce etc... and help to save the environment at the same time!
Most people are reluctant to acquire roaches because of the fear and misinformation that society in general has taught. We are born and bred to believe that roaches are carriers of disease and are filthy nasty little critters. Ever seen how much a cat grooms itself? Roaches put cats to shame, they are very clean insects. But you would only be able to observe this if you had them to start with. There are some pest species known to track through waste and then into contact with human food sources that can spread disease, but pest roach species are about 1% of the total number of roach species on this planet. Only a handful of them are actual pests. The roaches you would use as a feeder are tropical species that require very specific conditions in order to live and more importantly breed. My favorite saying to date seems to be "Unless you live in a mud hut in the middle of South America, these roaches will not infest your house"
I bid unto you, search the forums and compile together the previous posts outlining the benefits of roaches over crickets. They are still even more benefits than I listed, but if I keep going, this will turn into a full set of encyclopedias. If your parents have more specific concerns that no previous posts address, then feel free to ask the questions here on these forums. There are a lot of us members here that breed roaches and a couple that do it as a business, so plenty of knowledge to go around. If all else fails, get 1000 crickets, place them in a bin, feed/water them, then leave them be for two weeks. After that, ask your parents to open the lid to that bin themselves, the stink should help convince them.
