There is a couple of options using different paths to do what you want.
The first option I would suggest having a look at, is using Content Types and the Content Query Web Part to show you all tasks which are assigned to you ([Me]) within the current site collection. (This would not copy the item, but rather show all tasks within the site collection and this can then be filtered based on who is assigned to the task).
Display data from multiple lists with the Content Query Web Part -
https://support.office.com/en-in/article/Display-data-from-multiple-lists-with-the-Content-Query-Web-Part-d560f477-8fc1-4258-aeb2-70aa932e07da[
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The other options which could be used:
1. Have an Event Receiver which, when a new task is created, a copy is placed in the parent list. (C# / WSP option)
- How to: Create an Event Handler Feature -
https://msdn.microsoft.com/en-us/library/ms453149(v=office.12).aspx[
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- copy list items from one list to another in sharepoint -
http://stackoverflow.com/questions/1075323/copy-list-items-from-one-list-to-another-in-sharepoint[
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2. Create a custom workflow action (C#/WSP option), which takes the task and copies it to the parent list
- How to: Build and deploy workflow custom actions -
https://msdn.microsoft.com/en-us/library/office/jj163911.aspx[
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3. jQuery/SPServices to copy the task from child to parent site
- Create New List item by Copy Existing List Item and then edit the newly created List Item
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http://sharepointx-men.com/2013/03/13/part-i-create-new-list-item-by-copy-existing-list-item-and-then-edit-the-newly-created-list-item/[
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Hope this helps!
-DB
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