Try adding your parents as named drivers.
Please be aware of ghost brokers who offer low premiums but do not insure you, in reality they just steal your money, there are loads of them about, they even supply you with what looks like genuine paperwork. You only find out when you try to claim or are stopped by police.
I can only suggest you try comparison sites but sadly you are going to find it hard to find a reasonable quote, GOOD LUCK
Have you tried calling Adrianflux? They weren’t any good to me when i passed but i was trying to insure a z32 lol, They’re a insurance broker so might be able to find a company to get a good price. Also try calling insurers rather then going online, remember its the person on the phones job to get you signed up. 
This is good advice and more or less a must for young and inexperienced drivers.
I did the same for my daughter although she was in her late 20’s when she passed her test and took out her first insurance. Most companies were quoting £1800+ we finally found Admiral for around £800 but on a Picanto 1ltr. Best as mentioned to talk with them if you find they give a reasonably good quote online, I’ve since gone with them myself and I like a haggle on price they listened.
The cheapest Mk1 to insure will be the late model UK spec 1.6, after 1994. This model was created especially to push the MX5 down an insurance grouping. Footman James does a scheme for OC members. Also consider limited miles insurance.
A 30 year old MK1 MX5 might not be as reliable as you might think.
TBH, I don’t think a C1, which is basically a Toyota, is an unreliable as you think. It will be more reliable than a Mk1. I recall I had a first generation Aygo for a couple ofo weeks, while one of my 5s was being repainted. Not a lot of power, but a decent gearbox, and that 3 cylinder engine, provided you kept it on cam, could make decent progress. Fun little car if you work it. I think Toyota/PSA missed an opportunity with this car.
What’s your driving record/occupation, as the premium being quoted can’t just be down to age and postcode?
Hardtops don’t improve safety. They are a magnet for thieves (hardtops do get nicked).
And a correction, according to you profile, you are a member of the MX5 Owners Club, not “MX5 Fan Club”; if you are stating this, no wonder you are not able to secure a discount as a member.
Check the members’ area of the forum for deals, listing the companies that give a discount for club membership.