Your best option may be to do what I just did - get a Go Phone prepaid micro sim card from ATT. It comes out to 10 cents/minute for voice calls, but it's an easy way to get no contract service on your iphone. If you make a lot of phone calls, a traditional plan would probably be cheaper. But if you make relatively few calls, as I do (say 30-60 minutes/week) then this plan would be in the $12-24/month range for voice (for 30-60 min/week of calling). Plus they have roll over minutes as long as you renew before the current cycle expires. For example, I started mine with $25, that gives me 250 minutes that last 3 months. If after 2 months, I only use half that, I can renew a new cycle and have those 125 unused minutes carry over. Plus, not locked in to a contract.
I will say, the only reason I decided to go this route is because I'm having my regular/old phone # ported to Google Voice, and I'm planning on using GV over wifi on the iphone for the majority of my calling/texting (going to use a Clear mobile hotspot for on the go data). The prepaid SIM minutes are really a backup for when I don't have any data connection. I'm not sure if I would do this prepaid SIM if it was my only sort of coverage...really depends on how much time you spend on the phone.
Oh, they also now 'officially' offer data for iphone use on prepaid plans - again, it depends on your use. If you only use it sparingly to check email, weather, maps, etc, it's doable. But if you're streaming Youtube vids all day and downloading a ton of apps, mailing large attachments, etc, it's probably not a smart route to take. The prepaid data options are $5 for 10MB, $15 for 100MB, and $25 for 500MB. Because of my hotspot I don't plan on using the data options, but the great thing is - if I'm going on a road trip or camping or somewhere where I think I won't get signal on my hotspot, I can add 100 or 500MB at my leisure, so I'll be covered either way. Seems like a lot of hoops to jump through, but at the end of the day, I have a fully functioning iphone with unlimited everything for about $43, which takes into account the hotspot monthly fee and the SIM minutes. If I need to add some data to that, it would go up to $68/month (for the full $25 data option). If I just wanted to add a little data as a safety net I'd probably go with the $15 100MB option, which would make my bill that month $58. I doubt I'd go for the 10MB $5 option, because 10MB is a joke. Anyway, at most I'll be paying $68/month for unlimited everything, which is still cheaper than your basic restricted iphone plan.
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Just re-read your question and realized you weren't asking "how can we get this iphone to work" but rather "how can I get my parents' current SIM/plan on the iphone without data". Sorry...yeah, don't think you can, as was already said.