Finale is a complex program, and usually has more than one way to accomplish any task.  There is no right or wrong way, just different ways, some more preferable than others.  For placing text in a score and parts, some power users suggest using text expressions and creating an Expression Category for defining the placement parameters.  Make sure the category supports a "Score List", where you can define which staves will have the text expression, otherwise you will need to copy the text expression to each staff of the score where you want it to show on the part.  This will work, as will lyrics, which has also been suggested.  The Text Tool, IMHO the most logical tool for adding just text, can also be used.  Text blocks created with the text tool are of two varieties.  Those attached to pages, and those attached to measures.  Page attached text stays with the page and will show up on all parts as a matter of course.  This might be all you need.  Measure attached text will follow the measure no matter what page the measure finally appears on.  Page attached text is by default black, whereas measure attached text by default is red.  Expressions, BTW, by default are green.  Choose which variety from the bottom of the Text Tool drop down menu before placing the text.  It can also be modified after the fact in the text frame attribute dialog box.  As mentioned, if you are in "Scroll View", all new text blocks are measure attached.  Since measure attached text blocks do not support a "Score List" but only which single staff to place it on, found in the text frame attributes dialog box, you will have to copy the text to each staff of the score before it will show on the respective part staff.  Copying the text is fairly simple, if you use Finale's "Edit Filter" and define the filter to copy only measure attached text.  Then it is a simple matter to use the Select Tool to select the measure to which the text is attached, and drag/drop copy the measure, but only the attached text, to any other measure in the score.  Just to make sure all is well in the parts, you will probably want to re-generate all the parts from scratch.  This is done with "Document -> Manage Parts... -> Generate Parts".  The resulting dialog box is just a warning that any previously generated parts in the document will be replaced.  That is what you want, so go ahead.

 

I hope this clarifies any confusion.

 

[EDIT]  In addition, when entering expressions from the selection dialog box, there is the Assign button with a drop down arrow for more options, where you can assign the expression to the current staff, all staves, or pick/define an "Assignment List", not to be confused with the category "Staff List".  If the expression has already been entered, you can right click its handle to expose a pop up menu with "Assign to All Staves" (ALT-enter) or pick/create an Assignment List (CTRL-ALT-enter).  If you can remember the shortcuts, you can just select the expression handle and press the shortcut, and avoid opening the popup menu.