Now a caveat to this is that it may or may not work...
If you sit down with your parents calmly and talk to them about test strips vs the liquid tests. Set aside the inaccuracies of the strip tests.
They are worried about the cost of a liquid kit, and yes they are quite an investment....there is no denying that. There are different liquid test kits out there...API is the "go to" test kit but you have JBL, NT Labs, AquaCare, Hannah, TMC, Waterlife, Elos, Colombo....to name but a few.
It is a mind boggling minefield of different liquid testing kits and they stretch over a very wide budget and very deep pockets as well depending on which you choose.
As has been said, strips can be a bit unreliable. They are also easily contaminated (one shaky hand and the results go pearshaped) and their shelf life can also be questionable too.
Generally speaking even the most unshaken hand and careful aquarium owner can potentially go through more than a few packs of strips in a year...especially when first starting out or if the aquarium suffers an unexpected spike of some sort.
It can be just as expensive buying several packs of paper strips as one liquid test kit in a 12 month period. Cos with the best will in the world you cannot know that your aquarium will behave itself for the next 365 days in a row without experiencing a spike or some other water chemistry whoopsie that needs daily testing and water changes for days on end.
API is, as I said, the "go to" test kit but you don't really need to test for everything that it provides tests for. Slightly less expensive would be the NT Labs which you buy specific single type tests...ammonia, nitrate, pH etc - that could well work out being a cheaper option.
What you should do is with mum & dad, sit down and find and compare all of the liquid test kits available either in the local store or online...cost compare them with strips...compare their longevity of use.
You don't have to buy the most expensive liquid test kit out ther just cos its the one everyone else uses (cos I can bet most of those people with the API don't actually use every single available test in the box cos they aren't all necessary). Find one that suits your actual aquarium needs and budget...and you might be surprised that although expensive to buy initially, when comparing to strips and how long the shelf life and ease of use a cherrypicked liquid test kit like the NT Labs version might be the perfect fit for you
Don't be scared to look at other manufacturers of test kits (or anything to do with fishkeeping) you do not have to always by the brand that everyone else swears by, especially when on a limited budget or when you have to deal with justifying purchases with the Bank of Mum & Dad