You need points in both sides of the road to interpolate. In the process we will get also a buffer to clip the raster.
Create a buffer of your lines with a proper radius for a road (let's assume that 30m is a proper radius).
Dissolve the buffers.
Densify by interval the dissolved buffer, by a proper distance to get enough nodes (let's assume that 10m is a proper distance, I don't know the scale of your survey).
Extract vertices of the densified-dissolved-buffer.
You will have something like this:
Here comes the magic. We need to transfer the Z value from the survey points to the extracted points. Since QGIS 3.8, you can do it with the Join attributes by nearest tool. If you are using QGIS 3.4, let's take the opportunity to install a good plugin: refFunctions. You can install it from the Plugins menu.
refFunctions plugin adds some interesting functions to our expressions, under the Reference group.
In this case, we will use it to populate a new field of the extracted vertices with the value of one field (I called it Z) of the nearest point in the test_points layer.
Now we can make the interpolation with both layers:
The triangles remains outside the buffer, but that's innevitable.
Clip the raster by mask layer, and you will have a DEM to analize the slope of the road: