What
@Mark44 wrote is one usage of the term parameter as the numerical set-up of a situation. Here parameters mean the numeric frame, as room temperature in an experiment, or a certain range of tension which a screw has to tolerate if used normally. You can even consider ##(m,b)## in his example as parameters of lines, which means all pairs ##(m,b)## describe a set of lines and a certain pair fixes one line.
This leads to the other meaning (graphs). A parameter is a variable which accompanies a way along this graph. We say a curve is parameterized, which means a curve is a function ##f\, : \,[0,1] \longrightarrow \mathbb{R}## and ##t## is a parameter of the points ##(t,f(t))##, i.e. a walk of ##t## through ##[0,1]## corresponds to a walk through ##\{(t,f(t))\}##. It is not by chance, that those parameters are often noted as ##t##, because this reflects its association with time: like a walk in real life, where the parameter time on your watch corresponds to a certain location of your walk.
Other parameters can be temperature, pressure, distance, height or anyone dimensional quantity which serves as a measure. The term parameter is used, when the object of consideration is "walk" even if this walk is a metaphor like in the case of a drawn curve.
So the least common meaning is, that a parameter is a kind of measure, a numeric framework that describes a certain element in a set that contains all elements of a kind. That is a certain line ##(m_0,b_0)## in a set of lines ##\{(x,y)\,\vert \,y=mx+b\; ; \;m,b \in \mathbb{R}\,\}## as in Mark's example, or a certain point ##t_0## in the set of points on a line ##\{(x,y)\,\vert \,y(t)=mt+b\, ; \,t\in [0,1]\,\}## in my example. Therefore the confusion you've seen online might result from varying perspectives, i.e. sets of elements which are parameterized and as in our examples whether the lines are considered or the points on a line.
