Usually the funeral costs are just one of the final expenses that would get paid out of their estate. From a process point of view, it would be simpler for them to do nothing rather than give you money to give to a prepaid plan to give to the funeral home.
Funerals can be expensive, so one reasonable issue is cash flow right after death. In other words, the funeral home may want to be paid immediately, and it will take time for your partner to get named as executor and have access to their funds to pay for their funeral.
One way to handle the cash flow issue would be for you to pay for the funeral out of your pocket and then reimburse yourself from the estate. Another way would be for them to add your partner as POD (payable on death) or beneficiary to a bank or investment account that has enough to cover the funeral(s). A third way is for the funeral home to put the balance on a payment plan after the funeral - I'm sure funeral homes deal with the cash flow issue all the time.
I would not do an annuity. Annuities are pretty bad investment products and they're generally designed to provide a stream of income during life, not a lump sum at death.
If you could convince them, the best thing would be for them to invest the money themselves. Maybe, if it helps them feel better, they could open a separate account specifically designated for funeral costs, and then put whatever lump sum in that account they think they need. That would also make the beneficiary / POD approach much cleaner and simpler as well, since it would make sense to them that you would need to be added to that one account.
As far as investing safely goes, the joint life expectancy for a couple of 70 year old folks is surprisingly high. Another thing to keep in mind is that you probably won't have to deal with both funerals at once.
Good luck!