This was a very cute story. The characterisations were spot-on. I loved Severus' unwilling yet instinctive fascination with his son!
A bit of constructive criticism, though:
I did stumble in places where the grammar or wording was a little rough, almost enough to make me curious if English is not your native language. For example, you probably meant Harry 'clung' to Severus rather than 'clang' to Severus. Or, "Snape threw on the table a bunch of parchments." would be more accurate as "Snape threw a bunch of parchments onto the table."
I know from experience that it's hard to 'perfect' a story written on a deadline. The characterisation of this and the general story is very spot-on and fun to read. The grammar-roughness could easily be smoothed over with a grammar-beta.