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Hello this is Dr. Holloway but you can call me Dr. H for short. I am sorry to hear that your pet is having several bouts of sneezing. Chronic rhinitis (inflammation of the nasal passage) can be caused by several things including dust particles, foreign body that entered the nasal passage, fungi, etc. I would caution against using aspirin in a pet or any over the counter medication for that matter unless your pet has a specific disease process that deems its use (the only reason we would use aspiring is if your pet has bloodwork before-hand to rule out any underlying liver or kidney malfunction and aspirin was indicated to help with pain and inflammation, typically for joint pain).
But as an alternative you can try Benadryl. This would be more suitable for the rhinitis and possible allergies you have described. - Dr. H