Not that @mdsumner's answer is not perfectly correct, and honestly easier to understand, but why not just pull the areas from the "polygons" and "area" slot(s)?
sapply(slot(parcProj, "polygons"), function(i) slot(i, "area"))
I do notice a failure on both slot values and values returned by gArea in relation to holes (No Data area) verses real polygons. As illustrated by this code, gArea returns the correct area for a polygon with a hole whereas the area slot is not correct (as expected). However, when the "hole" is an actual polygon neither gArea or the area slot return the correct values. This hints at a bug where gArea is not returning the correct area with nested polygons.
library(rgeos)
library(sp)
Sr1 <- Polygon(cbind(c(2,4,4,1,2),c(2,3,5,4,2)))
Sr2 <- Polygon(cbind(c(5,4,2,5),c(2,3,2,2)))
Sr3 <- Polygon(cbind(c(4,4,5,10,4),c(5,3,2,5,5)))
Sr4 <- Polygon(cbind(c(5,6,6,5,5),c(4,4,3,3,4)), hole = TRUE)
Srs1 <- Polygons(list(Sr1), "s1")
Srs2 <- Polygons(list(Sr2), "s2")
Srs3 <- Polygons(list(Sr3, Sr4), "s3hole")
#### with hole
poly.hole <- SpatialPolygons(list(Srs1,Srs2,Srs3), 1:3)
sapply(slot(poly.hole, "polygons"), function(i) slot(i, "area"))
gArea(poly.hole, byid = TRUE)
#### without hole
Srs3 <- Polygons(list(Sr3), "s3")
poly.nohole <- SpatialPolygons(list(Srs1,Srs2,Srs3), 1:3)
sapply(slot(poly.nohole, "polygons"), function(i) slot(i, "area"))
gArea(poly.nohole, byid = TRUE)
#### with real polygon in place of hole
Sr4 <- Polygon(cbind(c(5,6,6,5,5),c(4,4,3,3,4)))
Srs4 <- Polygons(list(Sr4), "s4")
poly <- SpatialPolygons(list(Srs1,Srs2,Srs3,Srs4), 1:4)
sapply(slot(poly, "polygons"), function(i) slot(i, "area"))
gArea(poly, byid = TRUE)
# Plot data
par(mfrow=c(2,2))
plot(poly.hole, col=1:3, pbg="white");title("hole")
plot(poly.nohole, col=1:3);title("no hole")
plot(poly, col=1:4);title("real poly")