The days before Christmas are always exceptionally busy with people going to visit their families, etc.

24 is fully booked and 23 has only a few early or late trains with availability.

You will be refused boarding TGVs in Paris without reservation, they do carry out ticket checks in Paris terminals.

The few unreservable TER connections are your only option for the 24th with the single direct train at 08.36 being the quickest.

There may be a possibility of seats on some TGV re-appearing from cancellations.

On the 23rd, there's availability on the 20:24 from Paris Est. Be quick to book.

So since you need to travel to Salzburg on the same day I guess you want to maximize your stay in Strasbourg. Is it for the Christmas market ?

As everyone said trains will be extremely busy/sold out around that period. Here are a few options I’ve found :

Option 1 :

• TGV Paris-Est - Champagne-Ardenne TGV 08:05 - 08:45 20€
• TGV Champagne-Ardenne TGV - Strasbourg 09:34 - 11:05 20€

Option 2 :

• TGV Paris Gare de Lyon - Mulhouse 07:22 - 10:03 20€ (65 seats left so all good)
• TER Mulhouse - Strasbourg 10:46 - 11:39

Both options are good but do not wait to book of course. Questions welcome.

Thank you for the amazing recommendations 

I booked already, thanks alot.