You only need "for" in the first example to make it idiomatic:
Dad, can I use your car for the last time?
Dad, can I use your car one last time?
"The last time" can refer to a previous time, so without "for" to indicate a future possibility it doesn't fit with the tense of the question. There is another issue with this first example though - in at least some British dialects, adding "for the last time" to the end of a question can mean "this is the last time I am going to ask" and express annoyance.
That aside, the difference between the two questions as they are intended is that the definite article "the" makes it seem more final - "the last" is definitely the last.
"One last time" infers an additional use, and that perhaps the previous occasion was believed to be "the last".