2007 Mazda5 Transmission Issues

We have a used 2007 Mazda 5 automatic @ 210k KM (~130k miles).

We live in Canada and over the winter I noticed that on first driving the van in the cold, shifting from 1st gear to 2nd gear was acting odd. It would show on the dashboard as being in 2nd gear, but I could still hear the engine revving like it was still in 1st gear. Then it would 'hard shift' into 2nd gear probably a good 5 seconds or more after it had already shown as being in 2nd gear. This was really only on the first one or two times when shifting from 1st to 2nd and it seemed like it was less of an issue if I allowed the engine to warm up, more of an issue if I was driving it with a cold engine.

Does that make sense? I'm not a car person.

Now we are hitting a second issue and I'm wondering if the two are related. When I park the car and turn it off...It doesn't seem to actually go into park. So when I try to turn the car back on it won't start as it isn't picking up that the car is in park. This happened infrequently over the last year and would easily be fixed by shifting to drive and back to park, but now over the last week has happened multiple times and takes much longer to correct itself.

I've checked the transmission fluids and it appears to be full and the colour is a reddish colour.

Anyone familiar with any of these symptoms? Is my transmission dying?
2013 Mazda 5 Sport

I sometimes experience something similar to your first issue. It does happen only when the engine/tranny is not warmed up. However, it is not really a hard shift that happens, it's just the transmission hangs on to 1st gear a little longer than usual (1-2 secs) and then shifts to 2nd gear. As far as I remember, it only happens during the first shift. After that, it all normal.

Your two things could be related. It could be that the engine/tranny mounts need to be replaced and it could be affecting some sensor or something. I am not an expert, but it could be just your vehicle showing its age.
The two problems are usually unrelated. The no start in park issue is corrosion on the shifter pivot where the cable attaches at the transmission. The pivot is a $20 part at the dealer but they usually charge an hour to replace it. The 21mm nut that holds it in place is always tough to get off and you risk damaging the plastic cable end if you are too rough with it.
The other problem could be input/output shaft speed sensor related or possibly a bad TCM. Have it scanned with a good scan tool or the dealer tool to pick up any trans codes that might be lurking. If nothing, then you likely have a worn transmission.