My parents are both elders who have health problems they both took the Covid vaccine should I worry about the side effects ?

You must be right. The millions of vaccine receivers with no ill-effects have to be the anomalies.

There are several vaccines, you don't say which one so we can't help.
Lots of social media is simply spreading fear and nonsense, do your own research.
The CDC or WHO are both good sources of facts.

Elders are also far more likely to die if they catch COVID.

I would certainly prefer to try to prevent it instead of risking catching it.

Did they get the vaccine that needs a second dose - or are they done.

After 10 days - there are no side effects.  

Hank Aaron died of natural cases due to old age.  He was 86.  It is just pure chance that he happened to get the vaccine shortly before he died from SOMETHING ELSE.

(basically - he would have died without the vaccine - the vaccine didn't matter.)

37 Million here in the UK have had the vaccine, no one has died.

You would have noticed side effects by now, so I'm sure your parents are fine.  And don't believe all the stuff you read on fb.  Hank Aaron died of natural causes.

How long ago did they get the vaccine? They'd be able to tell you if they had any unreasonable side effects. Ask them.