Does it grow slowly or not?
if not:

delete_part(x+1, y);
create_part(-1, x+1, y, PT_GROW);
delete_part(x-1, y);
create_part(-1, x-1, y, PT_GROW);

if yes:

if(rand()%slow < 1){
delete_part(x+1, y);
create_part(-1, x+1, y, PT_GROW);
}
if(rand()%slow < 1){
delete_part(x-1, y);
create_part(-1, x-1, y, PT_GROW);
}

Again PT_GROW is the something that grows and slow the part on how slow it is. The first slow may be different from the other slow(means that it grows faster on a certain direction)

it's like a vine... it grows up things...

you don't want laser moving like photons?

If it is supposed to be a laser then find around line 667:
if(t==PT_ACID)
{
parts[i].life = 75;
}

put after that:

if(t==PT_GROW)
{
switch(rand()%2){
case 0:
parts[i].vx = -1;
case 1:
parts[i].vx = 1;
}
}

You could change 1 and -1 to a higher rate (It would be faster that way). But if it is very high, it would be unappealing to display.