Could someone tell me how strong the signals from the wireless devices are compared to those from lights, and perhaps those from the Sun as well?
At the surface of the Earth, the Sun delivers approximately 1 kW/m$^2$ [Wikipedia].
An average 100 W incandescent light bulb is only about 2.5% efficient so it emits 2.5 W of optical power (the rest is emitted as infrared or ultraviolet as described in the case of the Sun below).
Wireless routers emits about 0.1 W of power.
A cell phone emits about 1 W.
Let's put all that into a table, but let's make sure we include the frequency of the radiation emitted by each of these sources:
+----------------+-----------+-----------+
| Source | Power (W) | Frequency |
+----------------+-----------+-----------+
| Sun | 1000/m^2 | optical |
| Light bulb | 2.5 | optical |
| Cell phone | 1 | microwave |
| WiFi router | 0.1 | microwave |
| Microwave oven | 700 | microwave |
+----------------+-----------+-----------+
The Sun is by far the strongest emitter in our daily lives.
That's pretty obvious though if you think about the fact that looking at the Sun is painful and would destroy your eyes while looking at a WiFi router is no problem.
What is the scientific basis for claiming the radiation has or doesn't have effects on the human body?
There are two factors that determine whether radiation is hamful: flux and frequency.
Flux means roughly the number of photons flowing through a certain area per time.
Frequency means the frequency of each photon.
Power $P$ is related to frequency $\omega$ and flux $\Phi$ via
$$P = \Phi \hbar \omega \, .$$
However, power is not the only thing that determines harmfulness.
It turns out various materials have specific frequencies where they do and do not absorb radiation.
For example, glass does not absorb optical radiation, which is why you can see through it.
The Sun emits power over a range of frequencies but the peak is in the optical (i.e. visible) range.
That comes as no surprise because of course our eyes are evolved to see the radiation light that exists on Earth.
Optical radiation has relatively high energy and because of that it gets readily absorbed by the outer parts of your body (except for clear part of the eyes).
However, we're not usually exposed to enough optical radiation flux to do any harm.
For example, we don't usually encounter strong enough lights to burn us.
A really strong industrial laser would be a counterexample.
On the other hand, the part of the solar spectrum at frequencies just above the optical, known as "ultraviolet", has enough energy to damage your body cells, causing sunburn and skin cancer.
The part of the solar spectrum at frequencies below the optical, known as "infrared", is commonly called "heat".
The infrared is generally too low energy to destroy body cells at the levels coming from the Sun.
Incandescent light bulbs also emit a spectrum of radiation, and the story is relatively similar to the story we told for the Sun.
Now, cell phones, WiFi routers, and microwave ovens all produce microwave radiation, which in the range of 1 GHz frequency.
That's about 100,000 times lower frequency than visible light.
Microwave radiation penetrates your skin and goes through your body.
That's why microwave ovens work; the radiation permeates the food and heats it up.
Compare that to putting food right next to the heating element of a broiler in which case the food's outside cooks very quickly before the whole thing is done.
Anyway, the point is that microwave radiation penetrates your body.
That might sound scary, but microwave photons are too low in energy to damage your cells the way that ultraviolet does.
So even though microwaves heat you up a little bit, they don't give you cancer the way that sunlight does.
Now let's compare power levels in a cell phone and in a WiFi router.
Since the phone is right next to your head, about half of the phone's emitted power goes through your brain.
On the other hand, if you're $3\text{m}$ away from a WiFi router and we approximate the size of a head cross as $20\text{cm}$, then the fraction of the WiFi router's power going through your head is only about
$$\text{WiFi power fraction} = \frac{(20\text{cm})^2}{\underbrace{4 \pi (3\text{m})^2}_\text{surface area of sphere}} = 0.00035 \, .$$
So all together, the ratio of phone energy to WiFi router energy going through your brain is
\begin{align}
\frac{\text{phone power through brain}}{\text{WiFi power through brain}}
&= \frac{\text{phone emitted power}}{\text{WiFi emitted power}} \times \frac{\text{phone power fraction}}{\text{WiFi power fraction}} \\
&= \frac{1\text{W}}{0.1\text{W}} \times \frac{0.5}{0.00035} \\
&\approx 14,000 \, .
\end{align}
So a cell phone puts about 14,000 times more power through your brain than a WiFi router.
If your folks are afraid of the WiFi router, they should be terrified by cell phones.
