Is there direct train service from Lyon to Parma. Or must I change in Milan?

You would have to make more than one change and you are talking about at least 8-9 hours travel time. I suggest you look at budget flights (perhaps from Bologna).

Moved to Europe forum.

For European train schedules use bahn.de.

You can do this with one change at Milan, but it will take all day.

www.bahn.de/en says fastest Lyon-Parma train time is about 7.5 hours and yes 2 or 3 changes of train. So fly to Milan.

@PQ - your research is incomplete. See my post.The 11:38 requires only one change, but only runs between mid-July and September.

Well yes but the point is moot - too far for most to take train.

Flying to Milan seems to be the easiest way, but it can get pretty expensive. If you do not find any reasonable price for your date, you can also try flying to Bologna. The airline Hop! offers direct flights.

As others said, traveling by train could take pretty long, there is no direct train and it will not be cheap.

If you are on low-cost mode, you can have a look at buses as well, Flixbus and Baltour do this route, but you will need to change once anyway. In total it takes more than 9 hours.
Another option would be car-pooling, if there is any ride offered in platforms such as Blablacar. If you are lucky and somebody is offering a ride for your day, it should be cheaper and faster than a train or a bus.

I find the search engine Travelinho.com quite useful for this cases. It considers all transport modes and combines one segment with one transport mode and the next segment with another transport mode, so you would get also options such as to Milan by bus + car-pooling. It can be slow during the calculations though.

If you leave Lyon at 8.31, you may change at Torino PS and Milano Rogoredo. Arrival at Parma at 16.15.
For most other trains, you will have to change at Chambery too.
If you take the overnight train, you will have to change at Dijon and Brescia