How do I explain to my dad that a 1.5 ghz dual core is not a 3.0 ghz single core?

Hard to explain, but it's faster than single-core. This is probably an inaccurate way of demonstrating it:

Assume something gives 100% power (this would be single-core), now dual core would be more like:

100% + 50%.

So it's like half of what you previously had. So you are gaining more power and speed, it is BETTER than single core, but it's not double the power. Does that make sense? Also, if I am wrong, I would like to be corrected. That is all. I'm sure others here can either correct me, or explain it better. My analogy probably made very little sense as I'm not too experienced in comparing the two.

This is wrong to some extent.

A 1.5 GHz dual core will at best perform almost as well as a 3.0 GHz single core in some tasks. Almost because it requires some clock cycles to finish parallelism and such.
A 1.5 GHz dual core will at worst perform as well as a 1.5 GHz single core.

Sometimes you can split the work almost equally up, for example when summing numbers. If you have a list of numbers and split the list into two lists, ask the two cores to sum their lists up and then ask one of the cores to sum the sums, it would almost take the same amount of time. (For large lists you would notice no difference in speed.)

Sometimes you cannot split the work at all. Say that you want to calculate the nth fibonacci number. You won't be able to speed the process up because every fibonacci number is dependent of the earlier fibonacci numbers, and since you haven't calculated them yet, you're not able to speed this up. (With the formula F(n) = F(n-1) + F(n - 2))