Firstly, know that there are many interesting places in Parma to have kids' parties from the local parks (Bizzozero-Legambiente) to MagicoMondo (Bouncy castles) to bars (Federale Caffe') to nursery schools/baby "parkings" (Il Club dell'Orsetto Lavatore).
The www.parmakids.it website has more suggestions with detailed info (cost, contact info etc.)
Cake bakers delight: we've discovered two fantastic stores where you can purchase Wilton cake products, toppers, sprinkles, and all those other things you might be missing from home.
-Zanlari in Via Marchesi
-ArteDolce in San Pancrazio
For your paper goods and party decorations try:
-Carte Diem in V. La Spezia
-Parma Cart (but you will need a partita iva number to purchase here).
There are also magicians, entertainers and workshop-makers available to keep the kids busy.
-Il Club dell'Orsetto Lavatore
-L'Albero all'in Giu
and many others...
Lastly, arts & crafts stores include:
-Carte Diem (small selection but with items that are hard to find like googley eyes and mini pom-poms)
-Marchesi (via dei mille)
-Casamatti in Centro Torri.