Well firstly let me point out I'm sure this video isn't necessarily representative of how you usually play with Peach. I find it's very easy to make silly mistakes once a match starts to go downhill.
I think one main problem was you may not be that familiar with Zelda's attacks. Like at 2:23 you can see her charging your phantom, yet you still approach the ledge and get up about 3 seconds later, putting yourself in perfect line of sight for the attack despite having plenty of time to avoid it. A better idea would've been to use your parasol and float near the ledge and then quickly float to it after she uses it (she is forced to use it after a short amount of time), or quickly drop down from the ledge and then jump/parasol back up after the phantom is used. Hanging on the ledge isn't a great idea either because the phantom CAN hit you while you're on it (at least, I've been able to do that to people as Zelda).
The other problem is you are way too offensive and had very little defensive options. I don't think I saw you shield really, instead I only saw rolls. Shielding is definitely needed in this battle, especially any time you see Farore's Wind. If you see her begin to teleport, shield. See where she reappears and quickly punish if it's near you. Have some more patience, don't be in a rush to get back on the stage (that's what seemed to get you killed both times) or rush to get an attack on the opponent. The nice thing about Peach is you can stay off the stage for quite sometime and easily make it back to the ledge thanks to her float and her parasol.
I would practice, maybe with a friend or even a CPU, learning how to dodge her really predictable attacks from far away such as Din's fire, Farore's Wind, and Phantom Slash. Another good idea would be to train as Zelda. You might have won the match had you learned how to avoid those attacks (it was pretty even until she got you off the stage, then it went downhill).
And definitely take the advice of others mentioned. You already have a lot of the basics down so you can become an unbeatable Peach player if you take their advice. The Peach video thread is a GREAT resource. And there are lots of willing Peach players, I included, who are here to help with advice, 1 vs 1, if you need or want it..