Training a model for fall detection
I don't know of such a dataset, but I would suggest you to borrow ideas from mobile phone accelerometer projects. e.g.  the links below can be a good inspiration.

To proceed, I would record their daily movement and generate the normal movement thresholds (probably you could just use an off the shelf android phone utilizing its motion sensor).

Based on the collected dataset, you will define what are the normal movements (e.g. you can start with fixed rules). Later you dynamically adjust based on the feedback from your parents.

Concerning machine learning algorithms, After you collect high dimensional data for long enough time you can proceed with the algorithms you mentioned above. 

(e.g. setting it as a classification algorithm)

(speed, time, pt1, pt2, ...) -&gt; fall
(speed, time, pt1, pt2, ...) -&gt; no_fall


good luck!

[1] step detection https://github.com/danielmurray/adaptiv

[2] sensors overview https://developer.android.com/guide/topics/sensors/sensors_overview