Many US E46 323i and 323ci (1999-2000) had problems with their steptronic transmissions... and needed whole transmission replacement costing US$6000. The problem was due to BMW's claim of using "lifetime fill" transmission fluid that really doesn't last near a "lifetime". Eventually, this fluid breaks down causing transmission failure. This however usually happen around 100,000 miles.
With your parents car, because they bought it used, the transmission could have been badly abused by the previous owner. It's less common to see manual transmisssion failing compared to auto transmission; and with a relatively "mild" engine of the 316i, it makes it even less likely to fail.
Also, if the belts were to be worn so badly that they frayed, it means whoever who did the maintenance on the car did not do a good job or that your parents have skipped a service or two.
Get a 2nd opinion anyhow and move on from there. If a transmission is really needed, it can cost a lot less if you had procured one from a salvage yard instead.