When you ask the parrot a question, he will move in a certain direction, you have to follow him. He has to mention the direction where to go. Sometimes it seems illogical but it will work.You have to show your painting of the ultimate insult to the puppeteer.

I seem to know what to do but not the how. I have tried every conversation I can think of to get the puppeteer to run away by showing him the painting but so far have not managed the right combination.

Here's the step by step from UHS hints:
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How do I get the Guybrush and LeChuck puppets?
You take can't them when they're still on the puppeteer's hands. You need to get rid of the puppeteer somehow.
Scaring him would do the trick. You need to figure out some way to do this.
First, you need to talk to him somehow.
The LeChuck puppet will be unwilling to let you talk to anyone else. Asking him to do so won't do any good. Try something else.
Keep on asking him about Hellbeard until he gets mad and lets you talk to the Guybrush puppet.
Now, ask the Guybrush puppet to talk to the puppeteer. You'll have to ask twice.
Talk to the puppeteer until you discover something that would scare him.
You'll discover that he's scared of the Ultimate Insult. How can you use this fear to your advantage?
If he saw one, it'd probably scare him away.
You have a picture of one in your inventory. That would work.
Show the map/Ultimate Insult instructions to Hellbeard. (You may need to talk to the puppets again until Hellbeard pops up.)
Now, just pick up the puppets that he leaves behind.

Hope this gets you through!
MaG

Ah ha -- will follow your instructions and see how I do. Thanks ever so much.

Finally got the puppets - whewww! Now am stuck on the beach trying to get the correct conversation from both the pirate/the parrots. I just keep going around in circles with the same conversation over and over again.

With the pirate it has boiled down to three choices:

Ask why the Capt. is firing the cannon balls

Ask about the pirate's father

Tell pirate you are leaving.

I can't get anything about the Ult. Insult

And with the parrots - have discerned which is lying but again conversation just goes in circles -- never get the chance to ask about the bronze hat etc.

What am I doing wrong?!

Check this and it might help - Bronze hat Hints

It gives all the dialogues needed.

MaG

The hint list is excellent and I went clear back in my game to do Jambalya Island over again. So started anew in the Threepwood Inn. Tried talking to the tourist exactly as the hint list suggests but cannot get him to talk about the bronze hat -- he keeps saying he has no clue to that part of the Ultimate Insult. So am stuck again. The hint list claims one should talk about the bronze hat, then comment on the tourist's clothes, then ask about LaFeet. The tourist conversation I get is - he knows nothing about the bronze hat, never talks about Lafeet even though I talk about his (the tourist's clothes). I am missing something I guess but for the life of me do not know what.

Lilac, besides the two places Mags has given you, have you checked out The Spoiler Centre ? They have five walkthroughs for EFMI....

Jenny thanks for the extra WTs. I am afraid this game will drive me and all of you who are trying to help me batty. I seem to be doing all things the WTs talk about. Did find I was talking to the wrong tourist. Finally got that straightened out. Having done all the things I was supposed to but not in the order the WTs talk about -- I still cannot get the pirate on the beach to talk about the hat telling me he buried it under a boulder. So therefore cannot seem to get the conversation where the parrots tell me about the boulders. I am missing something or doing something wrong or not in order -- not sure what. I have even started over again re Jambalaya Island and tried following the one WT from MaG but still am unable to bring up the right conversation about the hat. Got the Monkey Mug with no problem. Maybe I should try for the Golden Man and see how I do then return to the Bronze Hat? Grrrrrrrrrrrrrrrrrrrrrr:(